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Figure 1
In the picture above, what is the difference in the areas of the two triangles?
Correct Answer: C
Look for similar triangles.
  1. First of all, we can use the Pythagorean theorem to find the third side of the smaller triangle. Since it is a right triangle with hypotenuse 10 and side length 6, we have:
    102 = 62 + x2
    where x is the length of the missing side. Solving for x, we see that x = 8. Thus the side lengths of the smaller triangle are 6, 8 and 10.
  2. Since the triangles both have two angles that have the same measure, their third angles are also of equal measure and the triangles are similar. Thus we can use proportions to find the missing side lengths.
  3. If we label the missing sides of the left triangle b for base and h for height, proportionality says that:
    Figure 2
  4. Cross multiplication equates these to 10h = 150 and 10b = 200, and thus h = 15 and b = 20.
  5. Since the area of a triangle is Area = (1/2)bh, we know that the area of the larger triangle is (1/2)(20)(15)=150 and the area of the smaller triangle is (1/2)(8)(6) = 24. Thus the difference in their areas is 150 – 24 = 126. The correct answer is C.

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