Problem Solving

In the diagram above, BF is an altitude drawn to the base AC, and AC is the side of square ACDE. If BF=h, and the area of triangle ABC equals Q, find the area of the square (shaded) in terms of h and Q.

A. hQ
B. q/h
C. 2*(Q/h)^2
D. 4*(Q/h)^2
E. 1/4*(Q/h)^2

The OA is D.

In this PS question, I know that the area of a triangle is equals to,

$$A\triangle=\frac{1}{2}bh,\ where,\ b\ is\ the\ base\ of\ triangle\ and\ h\ it\ altitude.$$

Then in this case, b=AC and h=BF, also I know that, $$\frac{1}{2}bh=Q$$

Now, how can I find the value of the side of the square in terms of Q and then find it area? Experts, I need your help with this PS question please! Thanks!

Hi AAPL,

I'll show you a couple of methods to do this. The first is a "quick and dirty" method applies a test scenario and seeing which answer choices match. The second method involves using geometrical theory and algebra to solve the question.

Empirical/testing method
Assume the square has sides of 4 (i.e. Area of square= 16)
Assume the altitude (BF) is 2, therefore, area of the triangle is 2*4/2 = 4, i.e. Q = 4.

Therefore, Q = 4 and h = 2 and we are looking for the overall area (16). Now, plug in Q and h into the answer choices and see what you get:

A. hQ = 2*4 = 8 ; not a match
B. q/h = 4/2 = 2 ; not a match
C. 2*(Q/h)^2 = 2* (4/2)^2 = 2*(2^2) = 2* (4) = 8 ; not a match
D. 4*(Q/h)^2 = 4* (4/2)^2 = 4*(2^2) = 4* (4) = 16 ; a match!
E. 1/4*(Q/h)^2 = 1/4* (4/2)^2 = 1/4*(2^2) = 1/4* (4) = 1 ; not a match

Caveat: You must check every single answer choice when using this method. If you get multiple matches, simple pick new numbers and test ONLY the answer choices that matched with the first numbers. One way of preventing this event is to strategically select your test values at the start.

In this case, we got lucky and the only match is OA D.

Geometry/Algebra
We know that BF = h and we can assume that the length of each side of the square is L
We also know that the area of the square is L^2

As you mentioned
Q = base * height *1/2 = L * h *(1/2)
Let's isolate the L:
2Q/h = L
We want L^2 to find the area, so: A = (2Q/h)^2 = (2^2) * (Q/h)^2 = 4*(Q/h)^2
So we have OA D.