A1 is a circle drawn about centre M. A2 is an arc with centre O. The drawing is not in any way to scale!
a) – Find the area of triangle OCD
Splitting the triangle into two right-angled triangles:
Height^2 = (17^2) – (15^2) = 64; so height = 8
Area = 0.5 * base * height
Area = 0.5 * 15 * 8 = 60
Therefore the area of the entire triangle is 60 * 2 = 120
b) – Find the angle COD in radians
Area = 0.5absinC
120 = 0.5 * 17 * 17 * sinC
sinC = 120 / (0.5 * 17 * 17)
sinC = 0.8304
C = 0.9799 rad
c) – Find the area of the shaded region R
Area of sector O – C – A2 – D – O = 0.5 * radius^2 * OCD
Area = 0.5 * 17^2 * 0.9799rad
Area = 141.5956
Area of semicircle A1 = 0.5 * pi * 15^2
Area = 353.4291
So, area of R = 353.4291 – 141.5956 = 211.83
Except that answer is wrong.
This question is in my Pure 1 maths textbook, and I was figuring it out yesterday. I kept getting the wrong answer and couldn’t understand why. Eventually Mum and I worked out that, in part b, sinC has 2 answers between 0 and 180 (or pi, if you prefer), and I got the wrong one. Had I used trig on the right-angled triangle to find half of OCD, like they did in the textbook explanation, I’d have come up with the correct answer. I didn’t, though. It’s quite annoying in that I didn’t actually do anything mathematically wrong; the only way I could realise something was wrong would be to think of the angle in degrees and then to realise that it would be 56deg. Were it 56deg then it would be nearly an equilateral triangle, which it isn’t. So it’s more of a logic thing than a mathematical one. There’s just no chance I’d realise this in an exam situation, though.
Can anyone think of any other reason I could have spotted my mistake?