Question

A circle has a chord that goes from `( pi)/3` to `(5 pi) / 12` radians on the circle. If the area of the circle is `7 pi`, what is the length of the chord?

Answer 1

≈ 2.02

Explanation:

I will attempt to explain the solution with no recourse to diagrams , although I recommend that you make sketches as we go.

If you draw the radii from the centre of the circle to both ends of the chord , then you will have a triangle.
Now the angle subtended at the centre by the chord will be.

`(5pi)/12 - pi/3 = pi/4`

The radius of the circle may be found , since we are given it's area, by using the following.

`pir^2 = 7pi →r^2 =( 7 cancel(pi))/cancel(pi) =7 → r = sqrt7`

The triangle now has 2 sides (radii) = `sqrt7`
with an angle between them of `pi/4`
and a third side - the chord.

To find the length of the chord , from the triangle, we need to use the `color(blue)" cosine rule "`

`a^2 = b^2 + c^2 - (2bc costheta )`

For this question a , is the chord , b and c , the radii and `theta = pi/4`

so `a^2 = (sqrt7)^2+(sqrt7)^2 -(2xxsqrt7xxsqrt7xxcos(pi/4))`

`= 7 + 7 - (14cos(pi/4)) = 14 - (9.899) = 4.101`

now `a^2 =4.101 rArr a " ( length of chord)" = sqrt4.101 ≈ 2.02`