A Circle with a radius R=8.17 has 2 congruent Tangent Lines attached to it. What is the area...

Question

There is a Circle, with a radius of R = 8.17. On the outside of the right of the circle is a point, directly right of the midpoint of the circle. 2 congruent lines leave from the point, and touch the circle at a tangent, on two sides, top and bottom. The area of contact is NOT the same area of the diameter chord, but some random chord in between. What the question wants is the shaded area in the space outside of the circle, but between the 2 tangent lines. The total distance from the edge of the left of the circle to the point at the rightmost is 3.5R. I have been trying a long time and cannot solve this. The closest I got was 80.662, but the answer is 7.56 * 10^1. Please help. I have drawn and uploaded an image below to make it easier to visualize.

 

http://postimg.org/image/vkx0lpojn/

Answer 1

draw 2 radii to the points of tangency at right angles

you have 2 right triangles

one leg is 8.17 and the hypotenuse is 28.595-8.17=20.425 where 28.595 is 3.5R

using the Pythagorean Theorem, the length of each tangent is 18.7198

using sin=opp/hyp

sin=18.7198/20.425

sin=0.9165141

using arcsin we get that the angle of the triangle to be 66.42167 degrees

double this to get the central angle of the arc of the tangents

this angle is 132.8433 degrees

find the area of this sector

the area of the whole circle is 209.6976

the area of the sector is (132.8/360)(209.6976)=77.357

now find the areas of the 2 right triangles

A=(1/2)(8.17)(18.7198) and double this answer

the area of both triangles together is 152.9407

now subtract the area of the sector to get the area that you want

152.9407-77.357=75.5837 which rounds to 75.6 which is what you said the answer is