CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 15 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 11:

Prove that parallelogram circumscribing a circle is a rhombus.

Given ABCD is a Parallelogram

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.

Two Tangents AP and AQ

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

Two Tangents TP and TQ Are Drawn to a Circle

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,