MathsICSE Class 10 · Tangent and Secant Properties

Tangent Perpendicular to the Radius and Equal Tangents from an External Point (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

The tangent at any point of a circle is perpendicular to the radius through that point. From a point outside a circle exactly two tangents can be drawn, and they are equal in length. So a tangent from $P$ has length $\sqrt{OP^2 - r^2}$, and the angle between the tangents is $180^\circ$ minus the angle at the centre.

Radii OA and OB meet the tangents at 90°, so the two right triangles are congruent and PA = PB, with length √(OP² − r²).

Illustration drawn for QuestionBuddy.

Understand Tangent–Radius Property and Equal Tangents

The CISCE syllabus for Tangent and Secant Properties begins with three facts; their proofs are to be taught.

1. Tangent ⟂ radius. The tangent at a point $A$ of a circle with centre $O$ is perpendicular to $O A$ .

*Why.* Every point of the tangent other than $A$ lies outside the circle, so it is farther from $O$ than $A$ is. $O A$ is therefore the shortest distance from $O$ to the tangent, and the shortest segment from a point to a line is the perpendicular.

2. Equal tangents. From an external point $P$ , two tangents $P A$ and $P B$ can be drawn, and $P A = P B$ .

*Proof.* In $△ O A P$ and $△ O B P$ : $∠ O A P = ∠ O B P = 9 0^{\circ}$ (tangent ⟂ radius), $O A = O B$ (radii), $O P$ is common. So the triangles are congruent (RHS) and $P A = P B$ . The congruence also gives $∠ A P O = ∠ B P O$ and $∠ A O P = ∠ B O P$ : $O P$ bisects both the angle between the tangents and the angle at the centre.

Consequences you will use again and again.

Tangent length: $P A = O P^{2} - r^{2}$ (Pythagoras in $△ O A P$ ).
**Quadrilateral $O A P B$ :** two right angles, so $∠ A P B + ∠ A O B = 18 0^{\circ}$ .
**Isosceles $△ P A B$ :** $∠ P A B = ∠ P B A = \frac{18 0 ^{\circ} - ∠ A P B}{2}$ , and $O P ⊥ A B$ .
Incircles: the tangent lengths from each vertex of a triangle (or quadrilateral) to its incircle are equal; in a right triangle with legs $a$ , $b$ and hypotenuse $c$ , the inradius is $r = \frac{a + b - c}{2}$ .
Tangents at the ends of a diameter are both perpendicular to that diameter, so they are parallel.

3. Touching circles. If two circles touch, the point of contact lies on the line joining their centres. For radii $R$ and $r$ , the distance between centres is $R + r$ if they touch externally and $R - r$ if they touch internally.

In the papers. 2024 Q9(i) could be done with the tangent–radius right angle and Pythagoras; 2024 Q3(ii) began with $∠ O QM = 9 0^{\circ} - 4 2^{\circ}$ ; and 2025 Q9(iii) began with $∠ B O X = 18 0^{\circ} - (9 0^{\circ} + 3 2^{\circ})$ .

Tangent–Radius Property and Equal Tangents: formulas to remember

Tangent ⟂ radius: $∠ O A P = 9 0^{\circ}$; $P A$ tangent at $A$ , $O$ the centre.
Length of a tangent: $P A = P B = O P^{2} - r^{2}$; From an external point $P$ .
Angle between tangents: $∠ A P B + ∠ A O B = 18 0^{\circ}$; $O A P B$ has two right angles.
Touching circles: $d = R + r (external), d = R - r (internal)$; Point of contact on the line of centres.
Inradius of a right triangle: $r = \frac{a + b - c}{2}$; From equal tangents; $c$ the hypotenuse.

How to solve Tangent–Radius Property and Equal Tangents questions

Draw the radius to every point of contact first; the right angles and equal tangents then do the work.

1Join the centre to each point of contact and mark the right angle.
2If two tangents come from one point, mark them equal and note the isosceles triangle they make with the chord of contact.
3For lengths, use Pythagoras in the right triangle formed by the radius, the tangent and the line to the centre.
4For angles, use $∠ A P B + ∠ A O B = 18 0^{\circ}$ and the base angles of the isosceles triangles.
5For touching circles, add or subtract the radii along the line of centres.
6Write a reason for each step, e.g. '(tangent ⟂ radius)', '(tangents from an external point are equal)'.

Tangent–Radius Property and Equal Tangents in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Equal tangents in a corner

Suppose a round table of radius 0.6 m is pushed into the corner of a room so that it touches both walls, which meet at right angles.

How far is each point of contact from the corner, and how far is the centre of the table from the corner?

Try it first, then show the solutionSolution

The radii to the two points of contact are perpendicular to the walls, so the centre, the two contact points and the corner form a square of side 0.6 m.
Each contact point is 0.6 m from the corner (equal tangents).
Centre to corner $= 0.6 2 \approx 0.85$ m (diagonal of the square).

Answer: 0.6 m to each contact point; about 0.85 m to the centre.

Memory hook: Radius to the wall is always square on to the wall.

Touching circles

Imagine two coins of radii 1.2 cm and 0.9 cm lying flat on a table and touching each other.

How far apart are their centres, and where is the point of contact?

Try it first, then show the solutionSolution

The point of contact lies on the line joining the centres.
Distance between centres $= 1.2 + 0.9 = 2.1$ cm (external touching).

Answer: 2.1 cm; the contact point is on the line of centres, 1.2 cm from the larger coin's centre.

Memory hook: Touching outside: add the radii.

Tricks to remember Tangent–Radius Property and Equal Tangents

Short memory aids. Say each one aloud once, then use it on a question the same day.

Radius meets tangent at 90
Every point of contact gets a radius and a little square.
Use it when: The first step of almost every tangent question.
Kite OAPB
Two tangents and two radii make a kite: two right angles, two pairs of equal sides, and $∠ P + ∠ O = 18 0^{\circ}$ .
Use it when: Angle questions with two tangents.
Ice-cream cone
Tangents from a point look like a cone: equal sides, and the line to the centre runs down the middle.
Use it when: Remembering $P A = P B$ and that $O P$ bisects $∠ A P B$ .

Tangent–Radius Property and Equal Tangents: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Radius from the tangent length · 1 mark · MCQBoard level

A point $P$ is 13 cm from the centre of a circle, and the tangent from $P$ to the circle is 12 cm long. The radius of the circle is:
(a) 1 cm
(b) 5 cm
(c) 7 cm
(d) 25 cm

Try it first, then show the solutionSolution

Tangent ⟂ radius, so $r = 1 3^{2} - 1 2^{2} = 25 = 5$ cm.

Answer: (b)

Example 2: Angles with two tangents · 3 marks · Short answerBoard level

$T A$ and $T B$ are tangents from $T$ to a circle with centre $O$ , and $∠ A O B = 11 2^{\circ}$ . Find $∠ A T B$ , $∠ T A B$ and $∠ O A B$ .

Try it first, then show the solutionSolution

$∠ O A T = ∠ O B T = 9 0^{\circ}$ (tangent ⟂ radius), so $∠ A T B = 36 0^{\circ} - 18 0^{\circ} - 11 2^{\circ} = 6 8^{\circ}$ .
$T A = T B$ (tangents from an external point), so $∠ T A B = \frac{18 0 ^{\circ} - 6 8 ^{\circ}}{2} = 5 6^{\circ}$ .
$∠ O A B = 9 0^{\circ} - 5 6^{\circ} = 3 4^{\circ}$ . (Check: $O A = O B$ gives $\frac{18 0 ^{\circ} - 11 2 ^{\circ}}{2} = 3 4^{\circ}$ ✓)

Answer: $∠ A T B = 6 8^{\circ}$ , $∠ T A B = 5 6^{\circ}$ , $∠ O A B = 3 4^{\circ}$

Example 3: Concentric circles · 3 marks · Short answerHarder board level

Two concentric circles have radii 13 cm and 5 cm. A chord $A B$ of the larger circle touches the smaller circle at $M$ . Find the length of $A B$ .

Try it first, then show the solutionSolution

$O M ⊥ A B$ (tangent ⟂ radius of the smaller circle), with $O M = 5$ cm.
The perpendicular from the centre bisects the chord, so $A M = M B$ .
$A M = 1 3^{2} - 5^{2} = 12$ cm, so $A B = 24$ cm.

Answer: 24 cm

Example 4: Three touching circles · 4 marks · Long answerHarder board level

Three circles with centres $A$ , $B$ and $C$ touch each other externally. $A B = 9$ cm, $B C = 11$ cm and $C A = 12$ cm. Find the three radii.

Try it first, then show the solutionSolution

Each point of contact lies on the line of centres, so $r_{A} + r_{B} = 9$ , $r_{B} + r_{C} = 11$ and $r_{C} + r_{A} = 12$ .
Adding: $2 (r_{A} + r_{B} + r_{C}) = 32$ , so $r_{A} + r_{B} + r_{C} = 16$ .
$r_{A} = 16 - 11 = 5$ cm, $r_{B} = 16 - 12 = 4$ cm, $r_{C} = 16 - 9 = 7$ cm.
Check: $5 + 4 = 9$ , $4 + 7 = 11$ , $7 + 5 = 12$ ✓

Answer: 5 cm, 4 cm and 7 cm

Example 5: Inradius of a right-angled triangle · 4 marks · Long answerHOTS / competency

A circle with centre $I$ is inscribed in $△ A B C$ , right-angled at $C$ , with $B C = 8$ cm, $C A = 15$ cm and $A B = 17$ cm. It touches $B C$ , $C A$ , $A B$ at $D$ , $E$ , $F$ .
(a) Prove that $I D C E$ is a square.
(b) Find the radius $r$ and the lengths $B F$ and $A F$ .

Try it first, then show the solutionSolution

(a) $∠ I D C = ∠ I E C = 9 0^{\circ}$ (tangent ⟂ radius) and $∠ C = 9 0^{\circ}$ , so $I D C E$ is a rectangle; $I D = I E = r$ , so it is a square and $C D = C E = r$ .
(b) $B F = B D = 8 - r$ and $A F = A E = 15 - r$ (equal tangents).
$A B = B F + A F$ : $17 = 23 - 2 r$ , so $r = 3$ cm.
$B F = 5$ cm and $A F = 12$ cm. Check by area: $\frac{1}{2} \times 8 \times 15 = 60 = r \times s = 3 \times 20$ ✓

Answer: (b) $r = 3$ cm, $B F = 5$ cm, $A F = 12$ cm

Example 6: Case study: how far to the horizon? · 4 marks · Case-basedHOTS / competency

Take the Earth as a sphere of radius 6400 km. An observer's eye is at $P$ , 0.2 km (200 m) above the ground on a hill top. The line of sight to the horizon touches the Earth at $H$ .
(a) Why is $∠ O H P = 9 0^{\circ}$ , where $O$ is the Earth's centre?
(b) Find the distance $P H$ to the nearest 0.1 km.
(c) Two horizon points $H_{1}$ and $H_{2}$ are seen in different directions. Compare $P H_{1}$ and $P H_{2}$ .

Try it first, then show the solutionSolution

(a) $P H$ is a tangent at $H$ and $O H$ is a radius, so they are perpendicular.
(b) $O P = 6400.2$ km. $P H^{2} = 6400. 2^{2} - 640 0^{2} = (0.2) (12800.2) = 2560.04$ , so $P H \approx 50.6$ km.
(c) Both are tangents from the same external point $P$ , so $P H_{1} = P H_{2}$ .

Answer: (b) About 50.6 km (c) Equal

Where students lose marks in Tangent–Radius Property and Equal Tangents

The slips examiners see most often, and what to write instead.

Mistake: Forgetting tangent ⟂ radius
Without the right angle, neither Pythagoras nor the angle sum can start.
Wrong working
$∠ O QM$ found without using $∠ O QP = 9 0^{\circ}$ .
Right working
$∠ O QM = 9 0^{\circ} - ∠ P QM$ .
Do this: Join the centre to each point of contact before anything else.
Usually costs: Reported in 2024 Q3(ii): candidates did not use the radius–tangent right angle, so $\angle OQM$ was wrong.
Mistake: Ignoring equal tangents
Two tangents from one point make an isosceles triangle with the chord of contact.
Wrong working
Treating $△ P S T$ as a general triangle when $P S$ and $P T$ are tangents from $P$ .
Right working
$P S = P T$ , so $∠ P S T = ∠ P T S = \frac{18 0 ^{\circ} - ∠ S P T}{2}$ .
Do this: Mark equal tangents with the same tick as soon as you see them.
Usually costs: Reported in 2024 Q1(xii): candidates missed that $\triangle PST$ is isosceles.
Mistake: Wrong distances with touching circles
Internal touching subtracts the radii; external touching adds them.
Wrong working
Radii 9 cm and 5 cm touching internally: centres 14 cm apart.
Right working
$9 - 5 = 4$ cm.
Do this: Sketch which circle is inside before adding or subtracting.
Usually costs: The whole numerical part.

Exam tips

2024 Q9(i) (3 marks): diameter produced to meet a tangent. The tangent–radius right angle gives $O P = 15$ and $P Q = 1 5^{2} - 9^{2} = 12$ cm, matching $P Q^{2} = P A \times P B$ .
2025 Q9(iii) (4 marks) started with $∠ B O X = 18 0^{\circ} - (9 0^{\circ} + 3 2^{\circ}) = 5 8^{\circ}$ ; the examiners note candidates who did not use the tangent–radius theorem or did not give reasons.
Both proofs (tangent ⟂ radius, equal tangents) can be asked; the RHS congruence proof is the usual one for equal tangents.
In the quadrilateral $O A P B$ , the angle at $P$ and the angle at $O$ always add to $18 0^{\circ}$ : a quick check.

Practice questions on Tangent–Radius Property and Equal Tangents

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
$T A$ and $T B$ are tangents from $T$ to a circle with centre $O$ . If $∠ A T B = 4 6^{\circ}$ , then $∠ A O B$ is:
(a) $4 6^{\circ}$
(b) $9 2^{\circ}$
(c) $13 4^{\circ}$
(d) $14 4^{\circ}$
Hint
$∠ A T B + ∠ A O B = 18 0^{\circ}$ .
Answer
(c)
Q2 · 2 marks · Short answerBoard level
A point $P$ is 29 cm from the centre of a circle of radius 20 cm. Find the length of the tangent from $P$ .
Hint
Tangent ⟂ radius.
Answer
$2 9^{2} - 2 0^{2} = 441 = 21$ cm
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The tangents drawn at the two ends of a diameter of a circle are parallel.
Reason (R): The tangent at any point of a circle is perpendicular to the radius through that point.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Both tangents are perpendicular to the same line.
Answer
(c)
Q4 · 3 marks · Short answerHarder board level
$T A$ and $T B$ are tangents from $T$ to a circle with centre $O$ , and $∠ T A B = 6 2^{\circ}$ . Find $∠ A T B$ , $∠ A O B$ and $∠ O A B$ .
Hint
$T A = T B$ .
Answer
$∠ A T B = 18 0^{\circ} - 12 4^{\circ} = 5 6^{\circ}$ ; $∠ A O B = 12 4^{\circ}$ ; $∠ O A B = 9 0^{\circ} - 6 2^{\circ} = 2 8^{\circ}$ .
Q5 · 3 marks · Short answerHarder board level
The incircle of $△ A B C$ touches $B C$ at $D$ , $C A$ at $E$ and $A B$ at $F$ . $B D = 6$ cm, $D C = 9$ cm and $A F = 4$ cm. Find $A B$ , $A C$ and the perimeter of the triangle.
Hint
$B F = B D$ , $C E = C D$ , $A E = A F$ .
Answer
$A B = 4 + 6 = 10$ cm, $A C = 4 + 9 = 13$ cm, $B C = 15$ cm; perimeter 38 cm.
Q6 · 2 marks · Short answerHarder board level
A chord of length 16 cm of a circle touches a concentric circle of radius 6 cm. Find the radius of the larger circle.
Hint
The radius of the smaller circle to the point of contact bisects the chord at right angles.
Answer
$8^{2} + 6^{2} = 10$ cm
Q7 · 4 marks · Long answerHOTS / competency
Two circles with radii 9 cm and 4 cm touch externally at $P$ . A common tangent touches the larger circle at $A$ and the smaller at $B$ (on the same side of the line of centres). The common tangent at $P$ meets $A B$ at $M$ .
(a) Prove that $M$ is the mid-point of $A B$ and that $∠ A P B = 9 0^{\circ}$ .
(b) Given that $A B = 12$ cm, find $M P$ .
Hint
$M A$ and $M P$ are tangents from $M$ to one circle; $M P$ and $M B$ to the other.
Answer
(a) $M A = M P$ and $M P = M B$ (tangents from an external point), so $M A = M B$ . Then $△ M A P$ and $△ M B P$ are isosceles; their base angles add to half of $18 0^{\circ}$ , so $∠ A P B = 9 0^{\circ}$ . (b) $M P = \frac{12}{2} = 6$ cm.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Angle between a tangent and the radius at the point of contact?Show answer
$9 0^{\circ}$
2. $O P = 10$ cm, radius 6 cm. Tangent length?Show answer
8 cm
3.Which congruence rule proves the tangents from a point are equal?Show answer
RHS
4.Radii 7 cm and 3 cm, touching internally. Distance between centres?Show answer
4 cm
5.Inradius of a right triangle with legs $a$ , $b$ and hypotenuse $c$ ?Show answer
$\frac{a + b - c}{2}$

Tangent–Radius Property and Equal Tangents: frequently asked questions

Why is a tangent perpendicular to the radius?

The radius to the point of contact is the shortest distance from the centre to the tangent, because every other point of the tangent is outside the circle. The shortest distance from a point to a line is the perpendicular.

How many tangents can be drawn from a point outside a circle?

Exactly two, and they are equal in length. From a point on the circle there is one tangent, and from a point inside there are none.

What happens when two circles touch?

The point of contact lies on the line joining the centres. The distance between the centres is the sum of the radii if they touch externally and the difference if they touch internally.