## In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that RTS is an acute angle.

To prove : RTS is an acute angle.Construction:Join RT and ST. Let RT intersect circle at point A. Join AS. Proof:Given RS is a diameter. O is the centre of the circle.Since RS is the diameter , RAS = 90˚ [Angle in semi circle is right angle]In ATS , RAS is an exterior angle and […]

## MRPN is cyclic, R=(5x – 13)°, N=(4x+4)°. Find measures of R and N.

Given R=(5x – 13)°, N=(4x+4)°Opposite angles of a cyclic quadrilateral are supplementary.R+N = 180˚5x -13+4x+4 = 1809x-9 = 1809x = 189x = 189/9x = 21R = 5x-13= 5×21-13= 92˚N=(4x+4)°N = 4×21+4= 84+4= 88N = 88˚Hence the measure of R = 92˚ and N = 88˚.

## In figure 3.57, PQRS is cyclic. side PQ side RQ. PSR = 110°, Find

(1)Given PSR = 110˚PQR = 180-110 = 70˚ [Opposite angles of a cyclic quadrilateral are supplementary.](2)PSR = ½ m(arc PQR) [The measure of an inscribed angle is half the measure of the arc intercepted by it]m(arc PQR ) = 2×PSRm(arc PQR ) = 2×110m(arc PQR ) = 220˚(3)Given side PQ side RQarc PQ arc RQ […]

## In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following.

(1)OA and OB are the radius of circle.Given AB = radius of circleAB = OA = OBOAB is an equilateral triangle.AOB = 60˚ [Angle of equilateral triangle = 60˚](2)ACB = ½ AOB [The measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle […]