Class 10 Maths: Circles – Extra Questions with Answers
Q. 1 A point $Q$ is located 15 cm from the center of a circle with a radius of 9 cm. Calculate the length of the tangent segment from point $Q$ to the circle.
Check Solution
Ans: C
Solution:
Let $O$ be the center of the circle, and let $T$ be the point of tangency on the circle. Then $\triangle OTQ$ is a right triangle with right angle at $T$. We have $OQ = 15$ cm and $OT = 9$ cm. By the Pythagorean theorem, $QT^2 + OT^2 = OQ^2$, so $QT^2 + 9^2 = 15^2$. Then $QT^2 = 15^2 – 9^2 = 225 – 81 = 144$. Thus, $QT = \sqrt{144} = 12$.
Answer: 12 cm
Q. 2 To draw a pair of tangents to a circle, which are inclined to each other at an angle of $60^\circ$, we have to draw tangents at the end points of those two radii, the angle between which is
Check Solution
Ans: B
Solution:
The radii to the points of tangency and the tangents form a quadrilateral. The sum of the angles in a quadrilateral is 360 degrees. The angle between the radii and each tangent is 90 degrees. Let x be the angle between the radii. Then, 90 + 90 + 60 + x = 360. So, x = 360 – 240 = 120.
Answer: 120
Q. 3 From a point B, the length of the tangent to a circle is 15 cm, and the distance of B from the center is 17 cm. What is the diameter of the circle?
Check Solution
Ans: A
Solution:
Let the circle’s center be O, the point of tangency be T, and the point B be the external point. We have a right triangle OTB, where OT is the radius (r), BT is the tangent (15 cm), and OB is the distance from B to the center (17 cm). Using the Pythagorean theorem: OT² + BT² = OB². So, r² + 15² = 17². Thus, r² + 225 = 289. Therefore, r² = 64, and r = 8 cm. The diameter is 2r = 2 * 8 = 16 cm.
Answer: 16 cm
Q. 4 The length of a tangent from a point B to a circle is 12 inches. The distance from point B to the center of the circle is 13 inches. What is the radius of the circle?
Check Solution
Ans: D
Solution:
Using the Pythagorean theorem, radius^2 + tangent^2 = distance^2
radius^2 + 12^2 = 13^2
radius^2 + 144 = 169
radius^2 = 25
radius = 5
Answer: 5 inches
Q. 5 A circle has a radius of 10 cm. A chord of the circle is tangent to a concentric circle with a radius of 6 cm. What is the length of the chord?
Check Solution
Ans: C
Solution:
Draw radii from the center to the endpoints of the chord (10 cm). Draw a radius to the point of tangency (6 cm), forming a right triangle. The chord is bisected by the radius to the point of tangency. Using the Pythagorean theorem: (chord/2)^2 + 6^2 = 10^2. (chord/2)^2 = 64. chord/2 = 8. chord = 16.
Answer: 16 cm
Q. 6 Construct the incircle of a $\triangle ABC$ with $AB = 5$ cm, $BC = 6$ cm, and $AC = 7$ cm. The steps involved in the construction, in sequential order, are: (A) Draw perpendicular $\overline{IM}$ from I onto $\overline{BC}$. (B) Taking I as the center and IM as the radius, draw a circle. (C) Draw $\triangle ABC$ with the given side lengths. (D) Draw angle bisectors of two angles, say $\angle B$ and $\angle C$, to intersect at I.
Check Solution
Ans: C
Solution:
Answer: C, D, A, B
Next Topic: Areas Related to Circles
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