Question

Please help me out please

Answer 1

The purple area is the sum of the circle sector and the triangle:

`A = A_s+A_t`

Let's compute them one at the time:

The sector is identified by an angle of 260°, because it is the remainder of a 100° angle.

We can build this simple proportion

`\text{total area}/\text{total angle}=\text{sector area}/\text{sector angle}`

The area of the circle is
`\pi r^2`, so we have

`\pi 8.35^2/ 360= A_s/ 260`

Solving for the sector area, we have

`A_s = (\pi\cdot8.35^2\cdot260)/(360) = (8.35^2\cdot 13\pi)/(18)`

The triangle is an isosceles triangle, because two of the sides are radii. This means that the height is also a bisector, so we can cut the triangle in two 90-50-40 triangles.

Using the law of sines, we can deduce that the height is

`8.35\sin(40)`

And half the base is

`8.35\sin(50)`

So, the area of the triangle is

`A_t = 8.35^2\sin(40)\sin(50)`

So, the purple area is

`A = A_s+A_t =\\(8.35^2\cdot 13\pi)/(18)+8.35^2\sin(40)\sin(50) = 8.35^2\left((13)/(18)\pi+\sin(40)\sin(50)\right)\approx 192.5`