# NCERT Mathematics Class 10 Exemplar Ch 9 Circles Part 5

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**EXERCISE 9.4**

1. If a hexagon ABCDEF circumscribe a circle, prove that .

2. Let s denote the semi-perimeter of a triangle ABC in which If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that .

3. From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If , find the perimeter of the triangle PCD.

Answer:

4. If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in Fig. 9.17. Prove that

5. Two circles with centres O and O’ of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O’P are tangents to the two circles. Find the length of the common chord PQ.

Answer:

6. In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC.

7. In Fig. 9.18, tangents PQ and PR are drawn to a circle such that . A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.

[Hint: Draw a line through Q and perpendicular to QP.]

Answer: 30°

8. AB is a diameter and AC is a chord of a circle with centre O such that The tangent at C intersects extended AB at a point D. Prove that .

9. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.