Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral
Solution:
Given, PQ and PR are the two tangents drawn from an external point to a circle with centre O.
We have to prove that QORP is a cyclic quadrilateral.
From the figure,
O is the centre of the circle
OR and OQ = radius of the circle
PR and PQ are the two tangents to the circle from an external point P.
We know that the radius of a circle is perpendicular to the tangent at the point of contact.
So, OR ⟂ PR and OQ ⟂ PQ
∠ORP = ∠OQP = 90°
We know that the sum of all interior angles in a quadrilateral is always equal to 360°
Considering quadrilateral PQOR,
∠OQP + ∠QOR + ∠ORP + ∠RPQ = 360°
90° + ∠QOR + 90° + ∠RPQ = 360°
180° + ∠QOR + ∠RPQ = 360°
∠QOR + ∠RPQ = 360° - 180°
So, ∠O + ∠P = 180°
Here opposite angles are supplementary.
Therefore, PQOR is a cyclic quadrilateral.
✦ Try This: From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.3 Problem 2
Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral
Summary:
Two tangents PQ and PR are drawn from an external point to a circle with centre O. It is proven that QORP is a cyclic quadrilateral
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