If the ratio of the area of rhombus CEGD to that of the square ABCD is 3:5, then what is the ratio of the area of the shaded region to that of the square ABCD?
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Is the answer 16/25? If we call the length of the sides of the square and rhombus x, then we know the area of the square is x^2 and the area of the rhombus is xh, where h is the height of the rhombus. Given the ratio, we know that
(xh)/(x^2) = 3/5
h/x = 3/5
h=(3/5)x
From here, my approach was to find the area of the square that is unshaded (trapezoid CEFD). To do this we have one missing piece -- the length of EF, one of the bases of this trapezoid. However, we can use the Pythagorean Theorem to find this. Since we know h in terms of x, form a right triangle by drawing in a line from E perpendicular to the base CD. Then this new right triangle has one side (3/5)x and hypotenuse x. Thus, the other side, y, is
y^2+((3/5)x)^2 = x^2
y^2 = (16/25)x^2
y = (4/5)x
Therefore, the length of EF is x-(4/5)x = (1/5)x. The area of trapezoid CEFD can then be found
((x+(1/5)x)/2)*h = ((3/5)x)*(3/5)x = (9/25)x^2
Then, we know the area of the shaded region is x^2 - (9/25)x^2 = (16/25)(x^2)
The ratio of this to the area of the entire square is
(16/25)(x^2)/(x^2) = 16/25.
