If two tangents inclined at an angle of 90 are drawn to a circle of radius 5cm

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Question 6 Circles - Exercise 9.1

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Answer:

∵ OA and PA are the radius and the tangent respectively at contact point A of a circle

of radius OA = 3 cm. So, ∠PAO = 90°

In right angled ΔPOA,

\tan30^{\circ}=\frac{OA}{PA}\Rightarrow\frac{1}{\sqrt{3}}=\frac{3}{PA}

⇒ PA = 3\sqrt{3}\ cm

Also, tangents drawn from a common point are equal in length.

So, PA = PB = 3\sqrt{3}\ cm

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Solution:

Given, two tangents are drawn to a circle of radius 3 cm, inclined at an angle of 60°

We have to find the length of each tangent.

From the figure,

Let PA and PC be the tangents drawn to a circle

PA and PC inclined at 60°

So, ∠APC = 60°

We know that the tangents through an external point to a circle are equal.

So, PA = PC

In triangle OAP and triangle OCP,

PA = PC

OA = OC = radius of circle

OP = OP = common side

By SSS criterion, triangles OAP and OCP are similar,

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

So, ∠OAP = ∠OCP = 90°

Since OA = OC = radius

∠OAP = ∠OCP

∠APC = ∠OAP + ∠OCP

So, 2∠OAP = 60°

∠OAP = 60°/2

∠OAP = 30°

In triangle OAP,

OAP is a right triangle with A at right angle.

tan 30° = OA/AP

By trigonometric ratio of angles,

tan 30° = 1/√3

So, 1/√3 = 3/AP

AP = 3√3 cm

We know, AP = CP = 3√3 cm

Therefore, the length of each tangent is 3√3 cm.

✦ Try This: If two tangents inclined at an angle 60° are drawn to a circle of radius 5 cm, then length of each tangent is equal to

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

NCERT Exemplar Class 10 Maths Exercise 9.1 Problem 9

If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to a. 3√3/2 cm, b. 6 cm, c. 3 cm, d. 3√3 cm

Summary:

If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then length of each tangent is equal to 3√3 cm

☛ Related Questions:

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Question 9 If two tangents inclined at an angle 60∘ are drawn to a circle of radius 3 cm, then the length of each tangent isA 3/2√3 cmB 6 cmC 3 cmD 3 √3 cm

Solution

Let $P$ be an external point and a pair of tangents is drawn from point P and angle between these two tangents is ${60}^{\circ }$

Radius of the circle $=3cm$ Join OA and OP

Also, OP is a bisector line of $\angle$ APC

$\therefore \angle APO=\angle CPO={30}^{\circ }OA\perp AP$ Also, tangents at any point of a circle is perpendicular to the radius through the point of contact.

In right angled $\Delta OAP,$ we have

$\tan {30}^{\circ }=\frac{OA}{AP}=\frac{3}{AP}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{AP}$

$\Rightarrow AP=3\sqrt{3}cm$

$AP=CP=3\sqrt{3}cm$ [Tangents drawn from an external point are equal]

Hence, the length of each tangent is $3\sqrt{3}cm.$