Question

What is the area of the minor segment cut off a circle of radius 10 cm by a chord of length 12 cm?

Could you please show me the working out for this question?

The answer in the textbook is 16 sq cm.

Thanks!!!

Answer 1

Make a sketch showing the chord of 12 and the the two radii of 10.
I see an isosceles triangle. Draw an altitude from the centre to that chord, making two congruent right-angled triangles.
let the height be x, then x^2 + 6^2 = 10^2
x^2 = 100-36 = 64
x = √64 = 8
So the area of the large triangle, 10,10,12 is
(1/2) (12)(8) = 48 cm^2

We have to find the central angle of the sector.
Let each angle at the centre of the right-angled triangles be Ø
sinØ = 6/10 = .6
Ø = 36.87‡
and the central angle is 2Ø = 73.74°

area of whole circle = π(10)^2 = 100π
area of sector/100π = 73.74/360
area of sector = 64.35 cm^2

sooo, the segment is 64.35 - 48 = 16.35 cm^2

(I carried all decimals my calculator could hold and only rounded off the final answer.)