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,