Solutions to Geometry Questions

Several days ago, I made a post about two simple geometry questions. Here I present my solutions:

Question 1:

Given that CF = FD and that , what fraction of square ABCD does occupy?

Solution 1

Place the corners of the square at (0,0), (0,1), (1,1) and (1, 0). The equation of line BF can be given by y = 1-0.5x and the equation of line AE can be given by 2x. Equating these gives x = 0.4 and y = 0.8 The area of triangle FED is therefore 1 - 0.2 - 0.25 - 0.4 = 0.15 or 3/20

Question 2:

Arc IFJ is one-quarter of a circle with center H. Given that HJ = IH = 6, and that EFGH is a rectangle with EH + HG = 8, what is the perimeter of the shaded region?

Solution 2

HF is a radius, and also a diagonal of EFGH, so EG = HF = 6. Next EG+HG = 8, but IH+HJ = 12, so the IE+GJ = 4. Finally, Arc IFJ is one-quarter of a full circle or 3π, so the total area is the sum of these segments, namely: = 10 + 3π.