Ttigonometry

Question

a circle with a radius of 35.4cm is cut into two segments by a chord with a length of 41.8cm. Determine the arc length and area of the smaller segmen

Answer 1

Join the centre of the circle to each end of the chord to make an isosceles triangle with a base of 41.8cm and equal sides of length 35.4 cm.

Divide the triangle into two right triangles by drawing the perpendicular bisector from the vertex at the centre of the circle on to the base. If 2θ is the angle subtended by the chord at the centre of the circle then sinθ=20.9/35.4 where 20.9cm is half the base length. From this θ=arcsin(20.9/35.4)=0.6315 radians approx. So 2θ=1.2631 radians approx. The arc length is 2rθ where r is the radius, so arc length=44.71cm approx. The area is 2r²θ/2=r²θ=791.43 sq cm.