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The hypotenuse of a right triangle is 16 ft longer than the length of the shorter leg. If the area of this triangle is exactly 120 ft2, what is the length of the hypotenuse in feet?
Triangle Leg and Hypotenuse
This problem involves the most famous theorem in math, the Pythagorean Theorem. While we could set up an algebraic solution—call the short leg x, call the hypotenuse (x + 16), etc.—the algebra quickly becomes very complicated. A much much better approach here would be to use backsolving.
As a very general rule, it’s a good idea to start with the median value, or the middle value of the five choices. (We do this because if it isn’t our answer, we can determine whether we need to try a larger number or a smaller number, and reduce the overall work we have to do when backsolving.) Here, the middle value would be (C), hypotenuse = 40.
Suppose the hypotenuse is c = 40. Then, the short leg is a = 40 – 16 = 24. Now, we could use the Pythagorean Theorem, a2 + b2 = c2. Before you pull out your calculator, though, we will preview another trick that we will discuss in the Geometry section. Notice that the numbers 24 and 40 have a Greatest Common Factor (GCF) of 8. We know that 24 = (3)(8) and 40 = (5)(8). This is a 3-4-5 right triangle, all multiplied by 8! Without a calculation, we know that the missing side must be (4)(8) = 32. We know that
(32)(24) = (32)(12)
We don't have to complete that calculation: we can see the area will be more than 300, way too big. We know (C) can't be the answer.
Here's the real application of what's great about starting with the middle value: if a hypotenuse of 40 is too big, then certainly one of 64 or 80 would have to be too big as well. On the basis of one choice, we can eliminate (C), (D), and (E), all at once!!
Now, we are down to just (A) & (B). Let's try (A). Let's say the hypotenuse is c = 26. Then, the small leg is a = 26 – 16 = 10. Again, we could use the Pythagorean Theorem, but again, we will note that 26 = (2)(13) and 10 = (2)(5). Another important Pythagorean triplet to memorize is the 5-12-13 triplet—that's what we have here. The missing leg must be (2)(12) = 24—again, knowing the Pythagorean triplets saves us from doing a calculation. From the two legs, we can get the area: 10 is the height and 24 is the base. We know that
(24)(10) = (12)(10) = 120
Bingo! That's exactly the right area! This one works!
Answer = (A)
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.