Class 10 Mathematics – Circles | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: Think of a spinning bicycle wheel rolling along a road. The road touches the wheel at exactly one point — that is a tangent. The spoke from the centre to that touch-point always stands perpendicular to the road. This is the key property of a tangent to a circle, and it is used in engineering, architecture, and even the design of railway tracks.

⭕ Topic 1: What Is a Circle

A circle is a round closed shape.
All points are equal distance from center.
That equal distance is the radius.
The center is usually called O.

⭕ Topic 2: Tangent to a Circle

A tangent is a line that touches a circle.
It touches the circle at one point only.
That point is called point of contact.
The tangent never enters the circle.

⭕ Topic 3: Secant of a Circle

A secant cuts through the circle.
It touches the circle at two points.
It goes inside the circle and comes out.
A secant is different from a tangent.

⭕ Topic 4: Tangent at a Point

At any point on a circle, one tangent exists.
We can draw tangent at any point.
The point where it touches is special.
Only one line can be tangent there.

⭕ Topic 5: Number of Tangents from a Point

Point inside circle: zero tangents possible.
Point on circle: exactly one tangent.
Point outside circle: exactly two tangents.
This rule always works for every circle.

⭕ Topic 6: Tangent Is Perpendicular to Radius

The tangent makes 90° with the radius.
This is at the point of contact.
Radius and tangent are perpendicular.
This is a very important theorem.
We use this in many problems.

⭕ Topic 7: Length of Tangent from External Point

From outside, tangent length can be found.
We use Pythagoras theorem for this.
PA² = OP² − OA² gives the answer.
Here PA is tangent, OA is radius.
OP is the distance to the center.

⭕ Topic 8: Two Tangents from Same Point Are Equal

From an outside point, draw two tangents.
Both tangents have equal length.
If PA and PB are tangents, PA = PB.
This is a key theorem for circles.

⭕ Topic 9: Tangent-Radius Properties

OA is perpendicular to tangent at A.
OB is perpendicular to tangent at B.
Triangle OAP has a right angle at A.
Triangle OBP has a right angle at B.
These triangles are congruent (RHS rule).

⭕ Topic 10: Angle Between Two Tangents

Two tangents from P make an angle at P.
Call it angle APB (A, B are contact points).
Angle AOB + angle APB = 180°.
If angle APB = 60°, angle AOB = 120°.
OP bisects both angle APB and angle AOB.

🎨 Section 2 — Easy Illustrations

Picture 1: All the important parts of a circle.

Picture 2: The tangent always makes 90° with the radius.

Picture 3: Two tangents from point P are always equal in length.

Picture 4: Tangent count depends on where the point is.

Picture 5: Use this right triangle to find tangent length.

✏️ Section 3 — Worked Examples

✏️ Example 1: Find the Tangent Length

A tangent PA is drawn from point P to a circle with center O. If OP = 13 cm and radius = 5 cm, find PA.

1 The tangent is perpendicular to the radius at A.

2 So triangle OAP has a right angle at A.

3 PA² = OP² − OA²

4 PA² = 13² − 5² = 169 − 25 = 144

5 PA = √144 = 12 cm

🎯 Length of tangent PA = 12 cm

✏️ Example 2: Find the Angle at Center

Two tangents PA and PB are drawn from point P to a circle with center O. If angle APB = 70°, find angle AOB.

1 We know: angle APB + angle AOB = 180°

2 This is because OA ⊥ PA and OB ⊥ PB.

3 In quadrilateral OAPB, all angles add to 360°.

4 90° + 90° + 70° + angle AOB = 360°

5 angle AOB = 360° − 250° = 110°

🎯 Angle AOB = 110°

✏️ Example 3: Find the Distance to Center

PQ is a tangent to a circle with center O at point Q. If OQ = 7 cm and PQ = 24 cm, find OP.

1 OQ is the radius, PQ is the tangent.

2 Angle OQP = 90° (tangent ⊥ radius).

3 OP² = OQ² + PQ² (Pythagoras)

4 OP² = 7² + 24² = 49 + 576 = 625

5 OP = √625 = 25 cm

🎯 Distance OP = 25 cm

📊 Section 4 — Cheat Sheet

Property	Statement	Quick Tip
Tangent	Touches circle at one point	1 point only
Secant	Cuts circle at two points	Goes through
Tangent ⊥ Radius	Tangent makes 90° with radius	Always 90°
Equal Tangents	PA = PB from external point	Both equal
Tangent Length	PA² = OP² − OA²	Pythagoras
0 Tangents	Point inside circle	No touch
1 Tangent	Point on circle	On the edge
2 Tangents	Point outside circle	Two lines
Angle Sum	∠APB + ∠AOB = 180°	Supplementary
OP Bisects	OP bisects ∠APB and ∠AOB	Cuts in half

📄 Section 5 — Take-Home Handout

✍️ A. True or False

1. A tangent touches a circle at two points. ___
2. A tangent is perpendicular to the radius. ___
3. From outside, we can draw two tangents. ___
4. The two tangents from a point are unequal. ___
5. A secant cuts the circle at two points. ___

✍️ B. Fill in the Blanks

1. A tangent touches a circle at ________ point(s).
2. Tangent makes ________° with the radius.
3. From outside, tangent length = √(________)
4. Two tangents from same point are ________.
5. ∠APB + ∠AOB = ________°

✍️ C. Solve These

1. A tangent PA from point P to a circle (center O, radius 9 cm). If OP = 15 cm, find PA.

Show your work here

2. Two tangents from point P. If ∠APB = 50°, find ∠AOB.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Tangent	a. Cuts circle at 2 points
2. Secant	b. PA = PB
3. Equal tangents	c. Touches at 1 point
4. Tangent ⊥ radius	d. 90° angle

🎨 E. Draw and Label

Draw a circle with center O. Mark an external point P. Draw two tangents PA and PB. Label the right angles.

Draw your diagram here

✅ Detailed Solutions

Solution to Q1: Length of Tangent from External Point

Question: A tangent PA is drawn from an external point P to a circle with centre O and radius 9 cm. If OP = 15 cm, find the length of PA.

1 The radius OA is perpendicular to the tangent PA (radius ⊥ tangent at point of contact).

2 In right triangle OAP: OP² = OA² + PA² (Pythagoras theorem).

3 15² = 9² + PA²

4 225 = 81 + PA² ⇒ PA² = 144.

5 PA = √144 = 12 cm.

∴ Length of tangent PA = 12 cm

Solution to Q2: Angle Between Two Tangents

Question: Two tangents PA and PB are drawn from an external point P to a circle with centre O. If ∠APB = 50°, find ∠AOB.

1 OA ⊥ PA and OB ⊥ PB (radius is perpendicular to tangent).

2 In quadrilateral OAPB: sum of all angles = 360°.

3 ∠OAP + ∠OBP + ∠APB + ∠AOB = 360°.

4 90° + 90° + 50° + ∠AOB = 360°.

5 ∠AOB = 360° − 230° = 130°.

∴ ∠AOB = 130°

🃏 Section 6 — Revision Flashcards

👉

⭕

What is a tangent?

A line that touches a circle at exactly one point.

🔎

What is a secant?

A line that cuts through a circle at two points.

📐

What angle does tangent make with radius?

90°

Tangent is always perpendicular to the radius.

❓

How many tangents from a point outside?

And both tangents are equal in length.

💡

What is the tangent length formula?

PA² = OP² − OA²

PA = √(d² − r²)

🔧

If PA and PB are tangents, what is true?

PA = PB

Two tangents from same point are equal.

🎯

∠APB + ∠AOB = ?

180°

They are supplementary angles.

🔗

How many tangents from a point inside?

No tangent can be drawn from inside the circle.

Class 10 Mathematics — Circles

Original Visual Revision Notes — Different Learner Friendly

Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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⭕ Class 10 Mathematics — Circles Interactive Visual Revision Notes Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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