Unit 2: Cyber-Safety and Digital Hygiene

Updated Nov 2025 | CBSE Class IX (Code 165) | 20 Min Read

Core Topics Overview

Safe Browsing, Encryption (Caesar Cipher), IT Act 2000, Emerging Threats (Deepfakes), Identity Protection, Digital Wellness.

Architecture of Safe Web Browsing

Browsing initiates complex data exchanges. The browser sends requests to servers containing metadata. This metadata identifies the user and their behavior.

Tracking Mechanics

IP Addressing: Numerical labels assigned to devices. They reveal geographical location and ISP.
Cookies: Text files storing state. Third-party cookies track users across different domains for profiling.
Fingerprinting: Scripts query device configurations (screen size, fonts) to create a unique ID without cookies.

Defense Tools

HTTPS: Encrypts data transfer. The padlock icon indicates encryption and server authentication.
Incognito Mode: Prevents local history storage. Does not hide activity from ISPs or websites.
VPN: Creates an encrypted tunnel. Masks IP address from external observers.

Visualizer: HTTP vs HTTPS Data Path

Red dots represent unencrypted data packets. Green dots represent encrypted packets.

Encryption Mechanics: The Caesar Cipher

Modern encryption (like AES used in HTTPS) is complex, but it started with simple substitution. The Caesar Cipher shifts every letter by a fixed number.

Plain Text (Input) Shift Value (Key)

ENCRYPTED OUTPUT

KHOOR

Formula: E(x) = (x + n) mod 26

*Try typing your name. Notice how if the key (shift) is known, the message is easily broken. Modern encryption uses keys with trillions of possibilities.

Public Wi-Fi Protocols

Open networks (cafes, airports) are susceptible to Man-in-the-Middle (MitM) attacks where hackers intercept data between you and the router.

Avoid

Banking transactions, entering passwords, or shopping on public Wi-Fi.

Adopt

Use a VPN or mobile data (hotspot) for sensitive tasks. Turn off “Auto-Connect” features.

Daily Browser Hygiene Checklist

Clear Cache & Cookies: Regularly purge temporary files to remove tracking scripts and free up space.

Update Browser: Ensure automatic updates are ON. Outdated browsers have unpatched security holes.

Check Extensions: Audit installed extensions. Remove any that you don’t recognize or don’t use.

Pop-up Blocker: Keep this enabled to prevent malicious ads (Adware) from opening automatically.

Online Gaming Protocols

Multiplayer environments are prime targets for social engineering and cyberbullying.

Griefing

Players who deliberately irritate or harass others to ruin the game experience. Action: Mute and Report immediately.

Skin Scams

Phishing sites promising “Free Skins” or “V-Bucks”. Action: Never enter login details on third-party generator sites.

Voice Chat

Predators use voice chat to build trust with minors. Action: Use voice changers or restrict chat to “Friends Only”.

How Search Engines Work

Search engines don’t search the *live* web in real-time. They search their own index.

1. Crawling

Automated bots (“Spiders”) browse the web, following links from page to page.

2. Indexing

Organizing the content found during crawling into a massive database (The Index).

3. Ranking

Algorithms determine the most relevant results for your query (e.g., PageRank).

Browser Data: What are you deleting?

Data Type	Function	Privacy Risk
Cache	Stores images/scripts to load pages faster on next visit.	Low (Mostly performance data).
Cookies	Stores login status, preferences, and cart items.	High (Used for tracking/profiling).
History	List of URLs visited.	Medium (Reveals habits if device is shared).

Social Network Protocol

Social platforms require specific hygiene practices to prevent data leakage and identity cloning.

Profile Locking

Restrict visibility of posts and photos to “Friends Only”. Public profiles are easily scraped by bots for identity theft.

Geolocation Tags

Disable auto-tagging of location in photos. Real-time location sharing facilitates physical stalking.

Friend Acceptance

Verify identities offline before accepting. Fake accounts often use stolen photos to infiltrate friend networks.

2FA (Two-Factor Auth)

Enable SMS or App-based authentication. Even if a password is stolen, the account remains secure without the second code.

💳 Secure Payment Hygiene

CVV Secrecy: Never share the 3-digit code on the back of your card via phone/email.

OTP Alert: Bank employees will NEVER ask for your OTP. It is for your eyes only.

Virtual Keyboards: Use on-screen keyboards for banking to defeat Keyloggers.

Email Forensics: “From” vs “Reply-To”

Hackers can spoof the “From” name to look like your Principal or Boss. Always check the actual email header.

From: “Principal Desk” <admin@school.com> // Looks Legit

Reply-To: hacker123@freemail.com // The Trap!

Subject: Urgent Fee Payment

…Click here to pay…

Always hover over the sender name to reveal the actual email address.

Mobile App Safety: Permission Audits

Apps often ask for more data than they need. Always question the relevance of the permission.

🔦 Flashlight App asking for Contacts? DENY 🗺️ Map App asking for Location? ALLOW 🎮 Game asking for Microphone? CHECK CONTEXT

Digital Shredding: Why “Delete” isn’t enough

When you delete a file or format a drive, the data is not erased. The operating system simply marks the space as “available”. Specialized software can recover this data.

Standard Delete

Like removing the table of contents. The book pages (data) remain.

Secure Wipe (Shredding)

Overwrites the actual data with random 0s and 1s multiple times. Impossible to recover.

Identity Protection & Ethics

⚠ Danger Zone: Cyber Grooming

A predatory process where an adult builds an emotional bond with a child online to gain their trust for sexual abuse or exploitation.

The Tactics

Secrecy: “Don’t tell your parents, they won’t understand us.”
Gifts: Offering game skins, recharge packs, or money.
Flattery: Excessive compliments to boost ego.

The Defense

Zero Secrecy: Never keep online friends a secret from guardians.
Refuse Gifts: Nothing online is truly “free”.
Block & Report: Immediate disengagement.

Digital Footprint Analysis

Your digital footprint is permanent. It is categorized into two types:

INTENTIONAL

👣 Active Footprint

Data you knowingly submit or publish.

Posting photos on Instagram/Snapchat.
Sending emails or WhatsApp messages.
Filling out online surveys.

HIDDEN

🕵️ Passive Footprint

Data collected without your explicit action.

IP Address logging by websites.
Search history tracking by Google/Bing.
Device GPS location history.

Concept	Definition	Example
Privacy	Right of the individual to control personal data.	Setting an Instagram profile to “Private”.
Confidentiality	Duty of the receiver to protect trusted data.	School keeping grades secure.
Digital Footprint	Trail of data left by online activities.	Comments, likes, IP logs.

Social Engineering: Hacking the Human

Attacks that manipulate people into revealing confidential information rather than hacking software directly.

🚩 How to Spot a Fake Link

http://secure-bank.fake-site.com/login

The Trap: “secure-bank” is just a subdomain. It looks real but is meaningless.

The Truth: The actual domain is “fake-site.com”. Always read the domain right before the “.com” or “.in”.

Phishing

Fraudulent emails mimicking trusted sources (e.g., “Bank Update Required”) to steal credentials.

Vishing

“Voice Phishing”. Fraudulent phone calls posing as support staff asking for OTPs.

Smishing

Phishing via SMS text messages containing malicious links.

Cyber Stalking Explained

Cyber stalking involves the use of the internet to harass, intimidate, or cause fear to a specific individual. It differs from general spam due to its targeted and persistent nature.

Primary Indicators:

Unwanted Messaging GPS Monitoring Identity Spoofing Threats of Violence

Forms of Cyber Bullying

Flaming: Online fights using electronic messages with angry and vulgar language.
Exclusion: Intentionally leaving someone out of a group chat or online game to hurt them.
Masquerading: Creating a fake identity to harass someone anonymously.
Doxing: Publicly revealing private personal information (address, phone number) about an individual.

Key Sections of the IT Act, 2000

The Information Technology Act is the primary law in India dealing with cybercrime and electronic commerce.

Section 66C

Identity Theft

Punishment for fraudulently making use of the electronic signature, password or any other unique identification feature of any other person.

Section 66D

Personation

Punishment for cheating by personation (pretending to be someone else) using computer resources or communication devices.

Section 67

Obscenity

Punishment for publishing or transmitting obscene material in electronic form.

Section 43

Damages

Deals with civil liability (compensation) for damage to computers (viruses, denial of service, data theft) without permission.

Netiquette (Digital Etiquette) Basics

No Spamming: Avoid sending repeated, irrelevant messages.
Respect Copyright: Do not plagiarize content; give credit.
No Trolling: Avoid deliberately provoking others for reaction.
Empathy: Remember there is a human behind the screen.

Digital Ethics: Intellectual Property

Software Licensing Models

Freeware

Copyrighted software available at no cost for unlimited use, but source code is hidden (e.g., Skype, Chrome).

Shareware

Software provided free on a “trial basis”. Payment is required for continued use or full features (e.g., WinRAR).

FOSS / OSS

Free and Open Source Software. Users can view, modify, and redistribute the source code (e.g., Linux, Firefox, LibreOffice).

Understanding Creative Commons (CC) Symbols

CC BY (Attribution)

Must credit the author.

CC NC (Non-Commercial)

Personal use only. No profit.

CC ND (No Derivs)

Cannot edit/remix work.

CC SA (Share Alike)

Must license new work same way.

Digital Wellness & Ergonomics

Safety isn’t just about data; it’s about physical health.

The 20-20-20 Rule

Every 20 minutes, look at something 20 feet away for 20 seconds to prevent eye strain.

Text Neck

Hold devices at eye level. Looking down for long periods strains the cervical spine.

E-Waste: The Environmental Footprint

Electronic waste contains toxic materials like Lead and Mercury. Never throw old phones/batteries in the dustbin.

Action: Locate your nearest authorized “E-Waste Collection Center”.

Authentication Security

Password strength relies on entropy (randomness). Length often beats complexity.

Weak (Low Entropy) Tr0ub4dor&3

Predictable substitutions.

Strong (High Entropy) battery horse staple correct

Long, random phrases.

Malware Analysis

Emerging Threats 2.0

AI & QR Based

Deepfakes

Media (Video/Audio) generated by AI that convincingly replaces a person’s likeness or voice with another’s.

How to Spot:

Unnatural blinking patterns (too much or too little).
Lip-sync issues (audio doesn’t perfectly match mouth).
Inconsistent lighting or shadows on the face.

Quishing

“QR Code Phishing”. Attackers paste fake QR stickers over real ones (e.g., at parking meters) or send QRs via email to bypass filters.

Defense Protocol:

Inspect physical stickers for tampering/overlay.
Preview the URL before the site loads (most scanners allow this).
Never pay via a QR code sent in an unsolicited email.

DDoS Attack (Distributed Denial of Service)

An attack meant to crash a website by flooding it with traffic.

The “Traffic Jam” Analogy

Imagine 1,000 fake cars suddenly blocking a highway. The real cars (legitimate users) cannot get through to their destination (website).

The Pillars of Cyber Security: The CIA Triad

Confidentiality

Preventing unauthorized access (Encryption, Passwords).

Integrity

Ensuring data is not altered/tampered with (Checksums).

Availability

Ensuring data is accessible when needed (Backups, preventing DDoS).

The Digital Gatekeeper: Firewall

A Firewall is a network security system that monitors and controls incoming and outgoing traffic based on security rules.

Analogy

Like a security guard at a gated community. They check the ID of every visitor (data packet) against a list of allowed guests (rules).

Firewall vs Antivirus

Firewall: Filters traffic before it enters the network. (Prevention)
Antivirus: Scans files after they enter the system. (Detection/Cure)

Virus

Needs a host file. Requires user execution to replicate. Destructive payload. Often attached to .exe or .doc files.

Worm

Standalone program. Self-replicating via networks. Does not need user action. Consumes bandwidth.

Adware

Advertising software. Automatically renders ads to generate revenue. Often bundles Spyware to track habits.

Botnets (Zombie Army)

A network of infected computers controlled remotely by a hacker without the owners’ knowledge. These “zombies” are used to launch massive attacks (DDoS) or send spam.

Ransomware

Malware that encrypts user data, demanding payment (ransom) for the decryption key. Backup is the only robust defense.

The 3-2-1 Backup Rule:

Keep 3 copies of your data.
Store them on 2 different media types (e.g., Hard Drive + Cloud).
Keep 1 copy offsite (different physical location).

Trojan Horse

Malware disguised as legitimate software (e.g., a free game) to trick users into installing it. It creates “backdoors” for hackers.

How Antivirus Software Works

Signature Detection

Compares files against a massive database of known virus “signatures” (code snippets). Effective against known threats.

Heuristic Analysis

Scans for suspicious *behavior* (e.g., a file trying to delete system folders) rather than specific code. Effective against new, unknown threats (Zero-day).

System Diagnostics: Is your device infected?

Performance Lag: Significant slowdown in processing speed or boot time.
Pop-up Storms: Frequent ad windows appearing even when the browser is closed (Adware sign).

Browser Redirection: Search queries redirect to unknown or malicious search engines.
Disabled Security: Antivirus or Firewall turns off automatically and cannot be restarted.

Reporting Mechanisms (India)

Step 1: Evidence Collection

Take screenshots of chats, profiles, and URLs. Do not delete the content immediately.

Step 2: Platform Reporting

Use the “Report User” or “Report Post” button on the specific social media app.

Step 3: Legal Escalation

Visit cybercrime.gov.in (NCRP) or dial 1930 for financial fraud.

NCRP

cybercrime.gov.in: Central portal for reporting all cybercrimes. Allows anonymous reporting for specific offenses.

1930

Helpline 1930: Immediate reporting for financial fraud. Integrated with banks to freeze fraudulent fund transfers.

Review Questions

Q1: Does using “Incognito Mode” hide your identity from your Internet Service Provider (ISP)?

Q2: What is the primary difference between a Virus and a Worm?

Q3: Which legal act in India governs cyber-safety violations?

Q4: Why is “123456” considered a weak password even though it is 6 digits long?

Q5: What should you check before entering passwords on a website?

Q6: A flash light app asks for access to your contacts and messages. What should you do?

Q7: Why is “formatting” a pen drive not safe before selling it?

Q11: Is “Freeware” the same as “Free and Open Source Software” (FOSS)?

Gen	Core Technology	Characteristics
1st	Vacuum Tubes	Huge, generated heat, slow (ENIAC).
2nd	Transistors	Smaller, faster, more reliable.
3rd	Integrated Circuits (IC)	Keyboards/Monitors introduced.
4th	Microprocessors (VLSI)	Current PCs. GUI OS introduced.
5th	Artificial Intelligence	Voice recognition, Robotics, Learning.

Feature	RAM (Random Access Memory)	ROM (Read Only Memory)
Types	SRAM, DRAM	PROM, EPROM, EEPROM

Feature	HDD (Hard Disk Drive)	SSD (Solid State Drive)
Parts	Moving parts (Spinning platters)	No moving parts (Flash chips)
Speed	Slower read/write	Extremely fast
Durability	Fragile if dropped	Shock resistant

Feature	Impact (e.g., Dot Matrix)	Non-Impact (e.g., Laser, Inkjet)
Mechanism	Hammer strikes ribbon	Sprays ink or fuses toner
Noise	High (Noisy)	Low (Quiet/Silent)
Unique Ability	Can make Carbon Copies	High resolution photos

Type	Extensions	Usage
Image	.JPG, .PNG, .GIF	Photos, Transparent graphics, Animations.
Audio	.MP3, .WAV	Music tracks, Sound effects.
Video	.MP4, .MKV, .AVI	Movies, YouTube clips.

Category	Proprietary (Paid/Closed)	FOSS Alternative (Free/Open)
Office Suite	Microsoft Office	LibreOffice / OpenOffice
Photo Editing	Adobe Photoshop	GIMP (GNU Image Manipulation Program)
Operating System	Microsoft Windows / macOS	Linux (Ubuntu, Fedora)
Audio Editor	Adobe Audition	Audacity

Feature	Plagiarism	Copyright Infringement
Definition	Presenting another’s work as your own.	Unauthorized use of protected material.
Nature	Ethical offense. Academic dishonesty.	Legal offense. Civil and Criminal.
Fix	Proper citation.	Obtaining permission or license.
Does Credit Help?	Yes.	No. Crediting a movie director doesn’t let you upload the movie.

Tag	Description	Visual Result
<b>	Bold Text	Example
<i>	Italic Text	Example
<u>	Underline	Example
<sup>	Superscript (Powers)	X²
<sub>	Subscript (Chemical)	H₂O
<strike>	Strikethrough (Deprecated)	Example

Tag	Attributes	Usage
<table>	border, width, align, bgcolor	Defines the container. ‘align’ centers the whole table.
<tr>	align, valign, bgcolor	Applies style to the entire row.
<td>	align, valign, colspan, rowspan	‘align’ moves text left/right. ‘valign’ moves text top/middle/bottom.

Protocol	Full Form	Primary Function
HTTP	HyperText Transfer Protocol	Viewing websites (Port 80)
FTP	File Transfer Protocol	Uploading/Downloading files (Port 21)
SMTP	Simple Mail Transfer Protocol	Sending Emails
SSH	Secure Shell	Secure Remote Login

Term	Definition	Key Characteristic
Artificial Intelligence (AI)	Any technique enabling computers to mimic human intelligence.	Broad umbrella term.
Machine Learning (ML)	A subset of AI. Machines improve at tasks with experience (data).	Learn without explicit programming.
Deep Learning (DL)	A subset of ML. Uses neural networks to train itself on vast data.	Mimics the human brain (Neurons).

Feature	Congruence (Class 9)	Similarity (Class 10)
Definition	Same shape and same size	Same shape, not necessarily same size
Symbol	≅	~
Angles	Corresponding angles are equal.	Corresponding angles are equal.
Sides	Corresponding sides are equal.	Corresponding sides are in the same ratio (proportional).

Theorem	Common Name	Statement	Syllabus Status
Th 6.1	BPT (Thales Theorem)	If DE \|\| BC, then (AD / DB) = (AE / EC)	Proof Required
Th 6.2	Converse of BPT	If (AD / DB) = (AE / EC), then DE \|\| BC	Motivation Only
Th 6.3	AA (or AAA) Similarity	If Angle A = Angle D, Angle B = Angle E, then Triangle ABC ~ Triangle DEF	Motivation Only
Th 6.4	SSS Similarity	If (AB / DE) = (BC / EF) = (AC / DF), then Triangle ABC ~ Triangle DEF	Motivation Only
Th 6.5	SAS Similarity	If (AB / DE) = (AC / DF) and Angle A = Angle D, then Triangle ABC ~ Triangle DEF	Motivation Only
Th 6.6	Area Theorem	(ar(ABC) / ar(DEF)) = (AB / DE)²	Proof Deleted (Application only)
Th 6.8	Pythagoras Theorem	In a right triangle, a² + b² = c²	Proof Deleted (Application only)
Th 6.9	Converse of Pythagoras	If a² + b² = c², it’s a right triangle	Proof Deleted (Application only)

CBSE Class 10 Maths Chapter 10: Circles

This chapter has two main ideas. First, a tangent and a radius are perpendicular. Second, two tangents from one external point are equal in length. This page explains these theorems and shows how to solve problems with them.

Line and Circle Relationships

A line in a plane can have one of three relationships with a circle.

1. Non-Intersecting

The line does not touch the circle at all. (0 common points)

2. Secant

The line cuts through the circle at two distinct points.

3. Tangent

The line touches the circle at exactly one point. This point is the ‘point of contact’.

Number of Tangents from a Point

The number of tangents you can draw to a circle depends on where the point is located.

Case 1: Point Inside Circle

Any line through P will cut the circle at two points (a secant).
Result: 0 Tangents

Case 2: Point On Circle

There is only one line that touches the circle at this single point.
Result: 1 Tangent

Case 3: Point Outside Circle

There are exactly two lines from P that touch the circle. (Theorem 10.2).
Result: 2 Tangents

First Main Idea: Theorem 10.1

“The tangent at any point of a circle is perpendicular to the radius through the point of contact.”

This means if XY is a tangent at point P, and O is the center, the radius OP will form a 90-degree angle with the line XY. So, OP is perpendicular to XY.

Proof Deconstructed (Shortest Distance)

Given: A circle with center O and a tangent XY at point P.
Goal: Prove that OP is perpendicular to XY.
Logic: Take any other point Q on the line XY (not P). Join OQ.
Point Q must be outside the circle. If Q were inside, the line would be a secant, not a tangent (it would cross twice).
Because Q is outside, the line OQ must be longer than the radius OP. So, OQ > OP.
This is true for every point on the line XY except P.
Therefore, OP is the shortest possible distance from the center O to the line XY.
A known geometric fact is: the shortest distance from a point to a line is the perpendicular distance.
Result: OP is perpendicular to XY.

Second Main Idea: Theorem 10.2

“The lengths of tangents drawn from an external point to a circle are equal.”

If you pick a point P outside the circle and draw two tangents that touch the circle at points Q and R, then the lengths PQ and PR will be equal.

Proof Deconstructed (RHS Congruence)

Given: A circle (center O), an external point P, and two tangents PQ and PR.
Goal: Prove that PQ = PR.
Construction: Join OP, OQ, and OR.
This creates two right-angled triangles: Triangle OQP and Triangle ORP.
Why right-angled? From Theorem 10.1. OQ (radius) is perpendicular to PQ (tangent), and OR (radius) is perpendicular to PR (tangent).
Compare the triangles Triangle OQP and Triangle ORP:
- OQ = OR (Both are radii of the same circle).
- OP = OP (This side is common to both triangles).
- angle OQP = angle ORP = 90° (From Theorem 10.1).
The triangles are congruent by the Right Angle-Hypotenuse-Side (RHS) rule.
Result: Since Triangle OQP is congruent to Triangle ORP, their corresponding parts are equal. Therefore, PQ = PR.

Two Results from One Proof

This RHS proof is powerful because it gives two results for the price of one (from CPCT – Corresponding Parts of Congruent Triangles):

PQ = PR — The tangent lengths are equal. (Used for length and algebra problems).
angle OPQ = angle OPR — The line OP bisects the angle between the tangents. (Used for angle-chasing proofs).

Theorem 10.1 vs. Theorem 10.2

Feature	Theorem 10.1 (Tangent-Radius)	Theorem 10.2 (External Point)
Main Idea	A tangent and radius are perpendicular (form a 90° angle) at the point of contact.	Tangents from a single external point are equal in length.
Number of Tangents	Deals with one tangent at a time.	Deals with a pair of tangents from one point.
Proof Method	Indirect proof (using “shortest distance”).	Direct proof (using RHS triangle congruence).
Key Result	angle = 90°	Side PQ = Side PR
Common Use	To create right-angled triangles for using the Pythagorean theorem.	To set up algebraic equations (e.g., x+8 = 15) and for proofs involving isosceles triangles.

Interactive Explainer: Theorem 10.2

Drag the external point P (the blue dot) with your mouse or finger. Observe how the two tangent lengths, PA and PB, change but always remain equal to each other.

This demo uses geometry to find the tangent points from point P.

Problem Types in Exercise 10.2

The problems in the textbook are not random. They test specific applications of the theorems. Use the filters to see examples of each type.

Numerical

Example: Q1, Q2, Q3, Q7

These are direct applications. Q2 and Q10 (proof) show that the angle at the center and the angle between tangents are supplementary (add to 180°). Q7 is a numerical version of Example 1, combining Thm 10.1 with Class 9 chord properties.

Proofs

Example: Q4, Q5, Q9, Q10

These problems ask you to prove corollaries. Q4 (tangents at ends of a diameter are parallel) is a key result of Thm 10.1. Q9 (Prove angle AOB = 90°) is a clever proof using SSS congruence from Thm 10.2.

Polygon Problems

Example: Q8, Q11, Q13

This family of problems uses one strategy: apply Thm 10.2 to every vertex, then add the equations. Q8 (AB+CD = AD+BC) leads directly to Q11 (a circumscribing parallelogram must be a rhombus).

Pinnacle Problems

Example: Q12

This is the capstone problem. It requires you to find the area of the triangle in two different ways (Heron’s Formula, and 1/2 × base × height) and set them equal. It combines Thm 10.2 with complex algebra.

Problem-Solving Flowchart

Not sure which theorem to use? Follow this logic flow to decide how to approach a problem. This chart shows the general decision-making process.

This flowchart shows a simplified path. Some problems (like Q12) combine multiple methods.

Key Proof Breakdowns

Let’s deconstruct the logic for two common exam proofs from Exercise 10.2.

Proof Breakdown: Q9 (Prove angle AOB = 90°)

Given: Two parallel tangents XY and X’Y’, and a third tangent AB at point C.

Logic Step 1: Look at the external point A. Tangents AP and AC are drawn.
Result (Thm 10.2): Triangle OPA is congruent to Triangle OCA (by SSS or RHS).
Key Takeaway: angle POA = angle COA. Let’s call this ‘x’. So angle POC = 2x.

Logic Step 2: Look at the external point B. Tangents BQ and BC are drawn.
Result (Thm 10.2): Triangle OQB is congruent to Triangle OCB (by SSS or RHS).
Key Takeaway: angle QOB = angle COB. Let’s call this ‘y’. So angle QOC = 2y.

Logic Step 3: Line POQ is a straight line (diameter) because the tangents are parallel.
Result: The angle on the straight line is 180°.
Equation: angle POC + angle QOC = 180°

Final Step: Substitute from Steps 1 and 2.
(2x) + (2y) = 180° -> 2(x + y) = 180° -> x + y = 90°
Conclusion: Since angle AOB = angle COA + angle COB = x + y, we have angle AOB = 90°.

Proof Breakdown: Q13 (Opposite sides… supplementary)

Given: A quadrilateral ABCD circumscribing a circle (center O).

Logic Step 1: This is the same logic as Q9, but applied four times.
At Vertex A: angle 1 = angle 2
At Vertex B: angle 3 = angle 4
At Vertex C: angle 5 = angle 6
At Vertex D: angle 7 = angle 8

Logic Step 2: The sum of all angles around the center point O is 360°.
Equation: angle 1 + angle 2 + angle 3 + angle 4 + angle 5 + angle 6 + angle 7 + angle 8 = 360°

Logic Step 3: Substitute using the pairs from Step 1.
( angle 2 + angle 2 ) + ( angle 3 + angle 3 ) + ( angle 6 + angle 6 ) + ( angle 7 + angle 7 ) = 360°
Simplified: 2(angle 2 + angle 3 + angle 6 + angle 7) = 360°
Result: angle 2 + angle 3 + angle 6 + angle 7 = 180°

Final Step: Group the angles by the opposite sides.
Angle for side AB: angle AOB = angle 2 + angle 3
Angle for side CD: angle COD = angle 6 + angle 7
Conclusion: (angle 2 + angle 3) + (angle 6 + angle 7) = 180° -> angle AOB + angle COD = 180°.

A Monograph on the Geometric Properties of Tangents to a Circle

1.0 Introduction and Foundational Concepts

In the study of Euclidean geometry, the relationship between a line and a circle stands as a cornerstone of geometric inquiry. Understanding the ways in which these fundamental figures can interact in a plane provides the basis for numerous theorems and applications. Of the possible configurations, the case of tangency—where a line touches a circle at a single point—is of particular importance, revealing deep and elegant properties of circular forms.

To begin our exploration, we must first establish a formal definition of a circle. A circle is the set of all points in a plane that are at a constant distance, known as the radius, from a fixed point, the centre. Given this definition, we can classify the three possible spatial relationships between a line and a circle within a shared plane:

Non-Intersecting Line: A line that has no point in common with the circle. It passes by the circle without making contact.
Secant: A line that intersects the circle at two distinct points. This line passes through the interior of the circle.
Tangent: A line that intersects, or touches, the circle at precisely one point. This line does not enter the circle’s interior.

The objective of this monograph is to systematically explore the properties, foundational theorems, and geometric propositions related to the tangent line, providing a comprehensive treatise on its significance.

2.0 The Nature of the Tangent

To fully appreciate the unique characteristics of a tangent, it is strategically important to view it not merely as a static definition but as a dynamic, limiting case of a secant. This perspective offers a more profound insight into its geometric properties and its relationship to the circle it touches.

The concept of a tangent can be deconstructed as a special case of a secant. Consider a secant line intersecting a circle at two points, P and Q. If we fix point P and rotate the secant line around it, the second point of intersection, Q, will move along the circumference of the circle. As the line rotates, Q will approach P. The limiting position of the secant, at the exact moment when point Q coincides with P, is the tangent line at point P. In this view, the tangent is the result of a secant whose corresponding chord length has diminished to zero.

This same principle can be observed from a different perspective by considering a series of parallel secants. If one draws a secant and then a series of lines parallel to it, the chords formed by the intersection of these lines with the circle will gradually decrease in length. Eventually, there will be two positions, one on each side of the original secant, where the length of the chord becomes zero. At these positions, the secant lines become tangent to the circle.

From this analysis, we can derive several key definitions and principles:

Uniqueness: At any given point on a circle, there exists one and only one tangent line.
Point of Contact: The single common point shared by a tangent line and a circle is formally termed the point of contact. The tangent is said to touch the circle at this point.
Historical Note: The term ‘tangent’ has its origins in the Latin word ‘tangere’, which means ‘to touch’. It was formally introduced into the mathematical lexicon by the Danish mathematician Thomas Fineke in 1583.

Having established a robust conceptual definition of the tangent, we now proceed to a rigorous examination of the foundational theorems that govern its behavior.

3.0 Core Theorems Governing Tangents

The true power of geometric analysis lies in its ability to formalize observed relationships into rigorously proven theorems. These theorems provide an unshakeable foundation upon which more complex propositions can be built. This section will present and prove the two central theorems that define the essential properties of tangents to a circle.

3.1 Theorem I: Perpendicularity of Radius to Tangent at the Point of Contact

This first theorem establishes the fundamental relationship between a tangent and the radius drawn to its point of contact.

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof:

Setup: Let there be a circle with centre O and a line XY that is tangent to the circle at a point P. The line segment OP is the radius to the point of contact. We aim to prove that OP is perpendicular to XY.
Core Argument: Take any point Q on the line XY, where Q is distinct from P. Since P is the only point the tangent shares with the circle, the point Q must lie outside the circle.
Implication: If Q lies outside the circle, the distance from the centre O to Q (OQ) must be greater than the radius of the circle (OP). Therefore, OQ > OP.
Conclusion: This inequality holds for every point Q on the line XY other than P. This demonstrates that OP is the shortest possible distance from the centre O to any point on the line XY. The shortest distance from a point to a line is, by definition, the perpendicular distance. Thus, OP is perpendicular to XY.

From this seminal theorem, two important corollaries arise:

Corollary 3.1.1: At any point on a circle, there can be one and only one tangent.
Corollary 3.1.2: The line containing the radius through the point of contact is defined as the normal to the circle at that point.

3.2 Theorem II: Equality of Tangent Lengths from an External Point

This second theorem addresses the case of two tangents drawn from a single external point and proves a crucial property regarding their lengths. First, we must define the term length of the tangent as the length of the line segment from the external point to the point of contact on the circle.

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

Proof using Right-Hand-Side (RHS) Congruence:

Setup: Consider a circle with centre O and a point P located outside the circle. Let PQ and PR be the two tangents from P to the circle, with Q and R as their respective points of contact. We must prove that PQ = PR.
Construction: Join the points OP, OQ, and OR to form the triangles Triangle OQP and Triangle ORP.
Identify Right Angles: From Theorem 10.1, we know that the radius is perpendicular to the tangent at the point of contact. Therefore, angle OQP and angle ORP are right angles.
Prove Congruence: In the right triangles Triangle OQP and Triangle ORP, we have:
- OQ = OR (Both are radii of the same circle).
- OP = OP (This side is common to both triangles).
Therefore, Triangle OQP is congruent to Triangle ORP by the Right-Hand-Side (RHS) congruence criterion.
Conclusion: Since the triangles are congruent, their corresponding parts must be equal. Thus, PQ = PR.

An Alternate Proof via Pythagorean Theorem

The same result can be derived using the Pythagorean Theorem. In the right triangle Triangle OQP: PQ² = OP² – OQ²

And in the right triangle Triangle ORP: PR² = OP² – OR²

Since OQ = OR (radii of the same circle), it follows that: PQ² = PR², which implies PQ = PR.

A further property can be concluded from the congruence of Triangle OQP and Triangle ORP: because angle OPQ = angle OPR, the line segment OP is the angle bisector of angle QPR. This means the centre of the circle always lies on the bisector of the angle formed between the two tangents.

4.0 Applications and Geometric Propositions

The true power and elegance of geometric theorems are revealed when they are applied to solve specific problems or establish new propositions. The core theorems governing tangents serve as foundational axioms from which a host of other geometric truths can be derived. This section reframes several illustrative examples as formal propositions to demonstrate the practical utility of these theorems.

4.1 Proposition 1: The Bisected Chord of Concentric Circles

Proposition: In two concentric circles, the chord of the larger circle which is tangent to the smaller circle is bisected at the point of contact.

Proof: Let C₁ be the larger circle and C₂ be the smaller circle, with a common centre O. Let AB be a chord of C₁ that is tangent to C₂ at point P. By joining O to P, we form the radius OP of the smaller circle C₂.

According to Theorem 10.1, the radius to the point of contact is perpendicular to the tangent. Therefore, OP is perpendicular to AB.

Now, considering the larger circle C₁, AB is a chord and OP is a line segment from the centre that is perpendicular to this chord. A fundamental property of circles states that a perpendicular drawn from the centre to a chord bisects the chord.

Therefore, AP = BP, and the chord AB is bisected at the point of contact P.

4.2 Proposition 2: The Angle Between Tangents and the Angle at the Center

Proposition: The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Proof: Let P be an external point from which two tangents, PQ and PR, are drawn to a circle with centre O. Here, Q and R are the points of contact. The angle between the tangents is angle QPR, and the angle subtended by the line segment QR at the centre is angle QOR.

By Theorem 10.1, the radii OQ and OR are perpendicular to the tangents PQ and PR, respectively. Thus, angle OQP = 90° and angle ORP = 90°.

The figure OQPR is a quadrilateral. The sum of the interior angles of a quadrilateral is 360°. Therefore: angle OQP + angle QPR + angle ORP + angle QOR = 360°

Substituting the known values of the right angles: 90° + angle QPR + 90° + angle QOR = 360° which simplifies to 180° + angle QPR + angle QOR = 360°

Solving for the sum of the remaining angles gives: angle QPR + angle QOR = 180°

Thus, the angle between the tangents and the angle at the centre are supplementary.

4.3 Proposition 3: Parallel Tangents at the Ends of a Diameter

Proposition: The tangents drawn at the ends of a diameter of a circle are parallel.

Proof: Let AB be a diameter of a circle with centre O. Let L₁ be the tangent to the circle at point A and L₂ be the tangent at point B. The diameter AB can be viewed as a straight line passing through the centre. The segments OA and OB are radii.

According to Theorem 10.1, the tangent is perpendicular to the radius at the point of contact. Therefore, L₁ is perpendicular to AB at point A, and L₂ is perpendicular to AB at point B.

Since both lines L₁ and L₂ are perpendicular to the same line AB, they must be parallel to each other. These propositions are direct and powerful extensions of the core theorems, illustrating how foundational principles are applied to deduce more complex geometric relationships.

5.0 Summary of Principles

This monograph has systematically explored the geometric properties of a tangent to a circle, from its conceptual definition to its core theorems and applications. The principal findings presented herein are consolidated below for clarity and retention.

A tangent to a circle is a line that intersects the circle at precisely one point. This unique point of intersection is formally termed the point of contact.
The first fundamental theorem states that a tangent is always perpendicular to the radius of the circle at the point of contact.
The second fundamental theorem establishes that the lengths of two tangents drawn to a circle from a common external point are equal.

Quick Q&A

Frequently Asked Questions (FAQs)

Is Chapter 10 important for board exams?

Yes. It is a high-weightage chapter. In the 2024-2025 board exam structure, the “Geometry” unit (which includes Circles and Triangles) has 22 marks. This single “Circles” chapter is often allotted around 8 marks, which may include MCQs and one 5-mark long-answer question.

Do I need to remember the Class 9 Circles chapter?

Yes, you need to know one theorem from Class 9: “The perpendicular from the centre of a circle to a chord bisects the chord.” This is used in Example 1 and Exercise 10.2, Q7 (the concentric circles problem).

What is the hardest problem in this chapter?

Many students find Exercise 10.2, Q12 to be the most difficult. This is the problem where a triangle circumscribes a circle of radius 4 cm, and you have to find the sides AB and AC. It requires using Theorem 10.2 (AE = AF = x) and then finding the triangle’s area by two different methods (Heron’s formula and 1/2 × base × height) and setting them equal.

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Author: Rezyo Staff Writer

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