Question

find r of the diagram.​

Answer 1

Answer:

Radius = 6cm.

Given.

A circle is circumscribed inside a quadrilateral.

SB = 5cm.

AR = 23cm.

AD = 29cm.

∠PDQ = 90°

OQ is the radius.

To Find.

OQ, the radius.

Solution.

We know that tangents from an external point to the circle are equal to one another. Using this theorem we get,

SB = BR

But SB = 5cm.

⇒ BR = SB = 5cm.

AQ = AR

AQ = 23 cm.

DQ = AD - AQ

DQ = 29 - 23

DQ = 6 cm.

But DQ = PD

⇒ DQ = PD = 90°

Join OP.

In quadrilateral OPDQ,

OP = OQ → Eq(1)(Radii of the same circle)

PD = DQ → Eq(2) (tangents from the same external point to the circle)

∠OPD = 90° → Eq(3)

∠OQD = 90° → Eq(4)

(Radius to the tangent at the point of contact is Perpendicular)

∠QDP = 90° → Eq(5) (Given)

∠QOP = 90° → Eq(6) (ASP of a quadrilateral)

From Equations 1, 2, 3, 4, 5 and 6, we can say that.

OPDQ is a rectangle.

i.e, OQ = PD

⇒ r = 6 cm. [PD = 6cm.]

Hence, the radius of the circle is 6cm.