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# If the lengths of the legs of a right triangle are integers, what is

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Senior Manager
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If the lengths of the legs of a right triangle are integers, what is  [#permalink]

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21 Sep 2019, 17:31
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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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Joined: 31 May 2018
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Location: United States
Concentration: Finance, Marketing
If the lengths of the legs of a right triangle are integers, what is  [#permalink]

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23 Sep 2019, 23:09
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

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Re: If the lengths of the legs of a right triangle are integers, what is  [#permalink]

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28 Sep 2019, 13:22
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.

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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Re: If the lengths of the legs of a right triangle are integers, what is  [#permalink]

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26 Dec 2019, 07:41
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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
AIDE: This is a huge piece of information. There aren't that 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 about triplets by some integer values.
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: A triangle has lengths 3, 4 and 5. In this case, the answer to the target question is the area = (3)(4)/2 = 6
Case b: A triangle has lengths 6, 8 and 10. In this case, the answer to the target question is the area = (6)(8)/2 = 24
Since we cannot 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 will always represent 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 to was telling us that the triangle must be a 3-4-5 right triangle.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: If the lengths of the legs of a right triangle are integers, what is   [#permalink] 26 Dec 2019, 07:41
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