Question

In the given figure, PQ and PR are two tangents drawn from an external point P to a circle with centre 'O'. If OQ = 3 cm and PS = 2 cm then, find the perimeter (in cm) of quadrilateral PQOR.
$\left[4\right]$

Answer 1

Given:
OQ = 3 cm
PS = 2 cm

OQ = OR = 3 cm [radius]
PO = PS + OS = 2 + 3 = 5 cm
$\left[1\right]$

By Theorem- The tangent at any point of a circle is perpendicular to the radius through the point of contact.

We know that PQ and PR are tangents, hence,
$\angle PQO=\angle PRO={90}^{\circ }$
$\left[0.5\right]$

$\text{Now, by applying Pythagoras}\text{theorem in}△OQP$

$P{O}^2=P{Q}^2+O{Q}^2$
$P{Q}^2=5^2-3^2$
$P{Q}^2=25-9=16$
$PQ=4cm$
​​​​​​​$\left[1\right]$

By Theorem - Tangents drawn from an external point to a circle are equal in length,
PQ = PR = 4 cm
​​​​​​​$\left[0.5\right]$

Perimeter of quadrilateral PQOR
= Sum of all sides
= PQ + PR + OQ + OR
= 4 + 4 + 3 + 3
= 14 cm
​​​​​​​​​​​​​​$\left[1\right]$