Brent@GMATPrepNow wrote: ↑Sat Mar 07, 2020 9:55 am
In the above diagram, the shaded square region is created by connecting each vertex to a midpoint. What fraction of square ABCD is shaded?
A) 1/8
B) 1/6
C) 1/5
D) 1/4
E) 1/3
Answer:
C
Notice that, within the diagram, we have four
identical right triangles (shaded below)
So, our strategy will be to assign a "nice" value to the sides of square ABCD, and then find the areas of the 4 shaded right triangles.
Let's say that
each side of square ABCD has length 2.
So, the area of a square ABCD = (2)(2) =
4
Now notice that we have 2
SIMILAR triangles hiding within our diagram.
Let's pull these two triangles out of the diagram to get the following:
ASIDE: Since we already let each side of square ABCD have length 2, we know that DC = 2 and FC = 1
Also, once we apply the Pythagorean theorem to this right triangle, we find that side DF has length
√5
To find the values of x and y, we'll use the fact that, if we have two similar triangles, then
the ratios of their corresponding sides will always be equal.
For example, we can write
√5/
2 =
2/
x
Cross multiply to get: (√5)(x) = (2)(2)
Solve: x =
4/√5
Likewise, to find the value of y, we can write:
√5/
2 =
1/
y
Solve to get: y =
2/√5
We now have:
Area of triangle = (base)(height)/2
So, the area of our shaded BLUE triangle = (
4/√5)(
2/√5)/2 =
4/5
This means the total area of all 4 shaded regions = (4)(
4/5) =
16/5
So, the area of the
shaded square =
4 -
16/5 = 4/5
So the
fraction of square ABCD that is shaded = (4/5)/
4 = 1/5
Answer: C
Cheers,
Brent
