Mathematics
In a triangle ABC, the incircle (center O) touches BC, CA and AB at points P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR;
given that ∠A = 60°.
Circles
6 Likes
Answer
ΔABC with its incircle having center O and touching BC, CA and AB at points P, Q and R, respectively is shown in the below figure:
(i) We know that,
The tangent at any point of a circle and the radius through this point are perpendicular to each other.
∴ ∠ORA = ∠OQA = 90°.
In quadrilateral AROQ,
∠ORA + ∠OQA + ∠QOR + ∠A = 360° [∵ Sum of interior angles in a quadrilateral = 360°]
⇒ 90° + 90° + ∠QOR + 60° = 360°
⇒ 240° + ∠QOR = 360°
⇒ ∠QOR = 360° - 240°
⇒ ∠QOR = 120°.
Hence, ∠QOR = 120°.
(ii) From figure,
Arc RQ subtends ∠ROQ at center and ∠QPR at the remaining part of the circle.
∴ ∠QPR = ∠QOR
⇒ ∠QPR = = 60°.
Hence, ∠QPR = 60°.
Answered By
3 Likes
Related Questions
ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with center O, has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.
In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate :
(i) ∠QOR,
(ii) ∠OQR,
(iii) ∠QSR.
In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.
Tangents AP and AQ are drawn to a circle, with center O, from an exterior point A. Prove that :
∠PAQ = 2∠OPQ