# 4. prove that the tangents drawn at the ends of a diameter of a circle are parallel.

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## Prove that the tangents drawn at the end of a diameter of a circle are parallel

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## Prove that the tangents drawn at the end of a diameter of a circle are parallel

Medium Open in App Solution Verified by Toppr To prove: PQ∣∣ RS

Given: A circle with centre O and diameter AB. Let PQ be the tangent at point A & Rs be the point B.

Proof: Since PQ is a tangent at point A.

OA⊥ PQ(Tangent at any point of circle is perpendicular to the radius through point of contact).

∠OQP=90 o …………(1) OB⊥ RS ∠OBS=90 o ……………(2) From (1) & (2) ∠OAP=∠OBS i.e., ∠BAP=∠ABS

for lines PQ & RS and transversal AB

∠BAP=∠ABS i.e., both alternate angles are equal.

So, lines are parallel.

$$\therefore PQ||RS.

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## Prove that the tangents drawn at the ends of a diameter of a circle are parallel

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.The tangent at any point of a circle is perpendicular to the radius through the point of contact.

## Prove that the tangents drawn at the ends of a diameter of a circle are parallel

**Solution:**

A tangent to a circle is a line that intersects the circle at only one point.

Let's draw the tangents PQ and RS to the circle at the ends of the diameter AB.

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

We know that radius is perpendicular to the tangent at the point of contact.

Thus, OA ⊥ PQ and OB ⊥ RS

Since the tangents are perpendicular to the radius,

∠PAO = 90°, ∠RBO = 90°

and ∠OAQ = 90°, ∠OBS = 90°

Here ∠OAQ is equal to ∠OBR and ∠PAO is equal to ∠OBS, which are two pairs of alternate interior angles.

If the alternate interior angles are equal, then lines PQ and RS should be parallel.

We know that PQ and RS are the tangents drawn to the circle at the ends of the diameter AB.

Hence, it is proved that tangents drawn at the ends of a diameter of a circle are parallel.

**☛ Check:**NCERT Solutions for Class 10 Maths Chapter 10

**Video Solution:**

## Prove that the tangents drawn at the ends of a diameter of a circle are parallel

Maths NCERT Solutions Class 10 Chapter 10 Exercise 10.2 Question 4

**Summary:**

Thus, we've proved that the tangents drawn at the ends of a diameter of a circle are parallel.

**☛ Related Questions:**

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

## Ex 10.2, 4

Ex 10.2,4 Prove that the tangents drawn at the ends of a diameter of a circle are parallel. Given: A circle with center O And diameter AB Let PQ be the tangent at point A & RS be the tangent at point B To prove: PQ ∥ RS Proof: Since PQ is a tangent at point A OA ⊥ PQ ∠ OAP = 90°

**Check sibling questions**

## Ex 10.2, 4 - Chapter 10 Class 10 Circles (Term 2)

Last updated at March 16, 2023 by Teachoo

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### Transcript

Ex 10.2,4 Prove that the tangents drawn at the ends of a diameter of a circle are parallel. Given: A circle with center O And diameter AB Let PQ be the tangent at point A & RS be the tangent at point B To prove: PQ ∥ RS Proof: Since PQ is a tangent at point A OA ⊥ PQ ∠ OAP = 90° Similarly, RS is a tangent at point B OB ⊥ RS ∠ OBS = 90° From (1) & (2) ∠ OAP = 90° & ∠ OBS = 90° Therefore ∠ OAP = ∠ OBS i.e. ∠ BAP = ∠ ABS For lines PQ & RS, and transversal AB ∠ BAP = ∠ ABS i.e. both alternate angles are equal So, lines are parallel ∴ PQ II RS

**Next**: Ex 10.2, 5 Important →

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### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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