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555-605 (Medium)|   Geometry|      
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gmatt1476
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If the lengths of the legs of a right triangle are integers, what is 21 Sep 2019, 18:31
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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
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B
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If the lengths of the legs of a right triangle are integers, what is Updated on: 16 Feb 2021, 19:38
Originally posted by BrentGMATPrepNow on 26 Dec 2019, 08:41.
Last edited by BrentGMATPrepNow on 16 Feb 2021, 19:38, edited 4 times in total.
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gmatt1476
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
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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 5
In 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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If the lengths of the legs of a right triangle are integers, what is 24 Sep 2019, 00:09
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If the lengths of the legs of a right triangle are integers, what is the area of the triangular region?

STATEMENT (1) The length of one leg is 3/4 the length of the other.
let the length of one leg = 4 than other = 3
area = 6
lenght of one leg = 8 other = 6
area = 24
INSUFFICIENT

STATEMENT (2) The length of the hypotenuse is 5.
--sides of the triangle are integers
hypotenuse = 5
the only possible integer values of legs are 3 and 4 (3,4, and 5 Pythagorean triplet)
--area of triangle = 6
SUFFICIENT

B is the answer
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If the lengths of the legs of a right triangle are integers, what is 28 Sep 2019, 14:22
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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.
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The triangle is a right triangle, so it must abide by the rules of Pythagorean's theorem: \(a^2\) + \(b^2\) = \(c^2\)

(1) There are an infinite amount of Pythagorean triples that have one leg equal to 3/4 of the other.

Side lengths could be: 3,4,5...... 6,8,10......9,12,15......12,16,20......etc

Not Sufficient

(2) The only right triangle that has a hypotnuse of 5, with all sides being integers, is the 3,4,5 Pythagorean triple.

Sufficient
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If the lengths of the legs of a right triangle are integers, what is 12 Nov 2020, 17:08
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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.

Many values fit this scenario; INSUFFICIENT.

(2) The length of the hypotenuse is 5.

Since we're told the lengths of the legs are integers, this legs must be 3 and 4. SUFFICIENT.

Answer is B.

Note that if the question did not state that the legs were integers, the answer would then be E.
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If the lengths of the legs of a right triangle are integers, what is 05 Apr 2024, 23:03
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