CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 15 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Question 11:

Prove that parallelogram circumscribing a circle is a rhombus.

Answer:

Given ABCD is a parallelogram in which all the sides touch a given circle

To prove: ABCD is a rhombus

Proof:

is a parallelogram

and

Again AP, AQ are tangents to the circle from the point A

Similarly,

Hence, parallelogram ABCD is a rhombus.

Question 12:

If two tangents are drawn to a circle from an external point, then

(i) They subtend equal angles at the center.

(ii) They are equally inclined to the segment joining the center to that point.

Answer:

Given on a circle , two tangents AP and AQ are drawn from an external point A.

To prove:

(i)

(ii)

Construction: Join , and

Proof: In

[Length of the tangents drawn from an external point]

[Radii of the same circle]

[common]

[By SSS theorem of congruence]

(i)

(ii)

Question 13:

Two tangents TP and TQ are drawn to a circle with center O from an external point T. Prove that

Answer:

Given A circle with center O and an external point T and two tangents TP and TQ to the circle, where P, Q are the points of contact.

To Prove:

Proof: Let

Since TP, TQ are tangents drawn from point T to the circle.

TPQ is an isosceles triangle

Since TP is a tangent to the circle at point of contact P

Thus,