gmatt1476 wrote:
If the lengths of the legs of a right triangle are integers, what is the area of the triangular region?
(1) The length of one leg is 3/4 the length of the other.
(2) The length of the hypotenuse is 5.
DS49302.01
Given: The lengths of the legs of a right triangle are integers ASIDE: This is a huge piece of information, since there aren't many right triangles that have
integer lengths.
Sets of 3 integers that could be the lengths of right triangles are called Pythagorean Triplets
Here are a few such triplets:
3-4-5, 5-12-13, 7-24-25, 8-15-17, etc
NOTE: It's important to remember that we can create additional triplets by multiplying each of the above triplets by some integer value.
For example, since 3-4-5 is a Pythagorean triplet, we also know that the following multiples will also be triplets:
6-8-10, 9-12-15, 12-16-20, 21-28-35, ....etc.Target question: What is the area of the triangular region? Statement 1: The length of one leg is 3/4 the length of the other. There are infinitely many right triangles that satisfy this condition. Here are two:
Case a: The triangle has lengths 3, 4 and 5. In this case, the answer to the target question is
the area \(= \frac{(3)(4)}{2}= 6\)Case b: The triangle has lengths 6, 8 and 10. In this case, the answer to the target question is
the area = \(= \frac{(6)(8)}{2}= 24\)Since we can't answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The length of the hypotenuse is 5In each Pythagorean triplet, the greatest value always represents the length of the hypotenuse.
When we examine all of the possible Pythagorean triples there are, only ONE is such that the greatest value is 5.
So, statement 2 is telling us that the triangle
must be a 3-4-5 right triangle, which means
its area \(= \frac{(3)(4)}{2}= 6\)Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
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