### NCERT Solutions for Class 10 Maths Chapter 10 Circle Exercise 10

**NCERT Solutions For Class 10 Maths Chapter 10 Circle**** Exercise 10**** **are prepared by specialised experienced mathematic teacher. Maths are most important subject of board and with the help of this chapter-wise NCERT solution and little practices you can get very good marks in your respective board exam. It also help students to build a foundation of upcoming class 11th and 12th. Student can also check the **Important Question with solution for class 9 to class 12.**

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**Class 10 Maths Chapter 10 Circle Exercise 10** contain only two exercises consists of 17 Problems and it covered the topic about a circle and various terms related to a circle such as a chord, segment, arc, etc **Check Previous chapter – NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry**

**Important Theorems –**

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

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.

### Exercise 10.1

**1. How many tangents can a circle have?**

**Answer:**

There can be **infinite** tangents to a circle. A circle is made up of infinite points which are at an equal distance from a point. Since there are infinite points on the circumference of a circle, infinite tangents can be drawn from them.

**2. Fill in the blanks:**

**(i) A tangent to a circle intersects it in …………… point(s).**

**(ii) A line intersecting a circle in two points is called a ………….**

**(iii) A circle can have …………… parallel tangents at the most.**

**(iv) The common point of a tangent to a circle and the circle is called …………**

**Answer:**

(i) A tangent to a circle intersects it in **one** point(s).

(ii) A line intersecting a circle in two points is called a **secant.**

(iii) A circle can have **two **parallel tangents at the most.

(iv) The common point of a tangent to a circle and the circle is called the **point of contact.**

**3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at**

**a point Q so that OQ = 12 cm. Length PQ is :**

**(A) 12 cm**

**(B) 13 cm**

**(C) 8.5 cm **

**(D) √119 cm**

Solution:

**Note:** PQ = √119

**Question 4.**

**Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.**

**Solution:**

### Exercise 10.2

**Question 1.**

**From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is**

**(a) 7 cm**

**(b) 12 cm**

**(c) 15 cm**

**(d) 24.5 cm**

**Solution:**

**Question 2.**

**In figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to**

**(a) 60°**

**(b) 70°**

**(c) 80°**

**(d) 90°**

Solution:

**Question 3.**

**If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to**

**(a) 50°**

**(b) 60°**

**(c) 70°**

**(d) 80°**

**Solution:**

**Question 4.**

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

**Solution:**

**Question 5.**

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

**Solution:**

**Question 6.**

**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.**

**Solution:**

**Question 7.**

**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.**

**Solution:**

**Question 8.**

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

Solution:

**Question 9.**

**In figure, XY and X’Y’ are two parallel tangents to a circle , x with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB = 90°.**

Solution:

**Question 10.**

**Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.**

**Solution:**

**Question 11.**

**Prove that the parallelogram circumscribing a circle is a rhombus.**

**Solution:**

**Question 12.**

**A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see figure). Find the sides AB and AC.**

Solution:

**Question 13.**

**Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.**

**Solution:**