In Fig. 9.5, AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to
a. 4 cm
b. 2 cm
c. 2√3 cm
d. 4√3 cm

Solution:

Given, AT is a tangent to the circle with centre O

The length of OT is 4 cm

Also, ∠OTA = 30°

We have to find the length of AT.

Join O and A.

OA will be the radius of the circle.

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

So, ∠OAT = 90°

In triangle AOT,

AOT is a right triangle with A at right angle.

By pythagorean theorem,

cos 30° = AT/OT

By using trigonometric ratios of angles,

cos 30° = √3/2

So, √3/2 = AT/4

On cross multiplication,

4√3/2 = AT
AT = 2√3 cm

Therefore, the measure of AT is 2√3 cm.

✦ Try This: In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA = 17 cm, then the length of AB (in cm) is

Given, AB and AC are tangents to a circle with centre O.

Radius of the circle = 8 cm

We have to find the length of AB if OA is 17 cm.

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

So, ∠OBA = ∠OCA = 90°

In triangle OBA,

OBA is a right triangle with B at right angle.

By pythagoras theorem,

OA² = AB² + OB²

(17)² = AB² + (8)²

289 = AB² + 64

AB² = 289 - 64

AB² = 225

Taking square root,

AB = 15 cm

Therefore, the length of AB is 15 cm.

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

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NCERT Exemplar Class 10 Maths Exercise 9.1 Problem 6

In Fig. 9.5, AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to a. 4 cm, b. 2 cm, c. 2√3 cm, d. 4√3 cm

Summary:

In Fig. 9.5, AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to 2√3 cm

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☛ Related Questions:

• In Fig. 9.6, if O is the centre of a circle, PQ is a chord and the tangent PR at P makes an angle of . . . .
• In Fig. 9.7, if PA and PB are tangents to the circle with centre O such that ∠APB = 50°, then ∠OAB i . . . .
• If two tangents inclined at an angle 60° are drawn to a circle of radius 3 cm, then length of each t . . . .