Geometry Question
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- MartyMurray
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The ratios of the measurements of an isosceles right triangle can be thought of in two main ways.
Side Side Hypotenuse
..1.....1......√2
√2/2 √2/2...1
Looking at the second way, you can see that each side is 1/2 x length of the hypotenuse x √2.
Compare that relationship to the numbers given, 16√2 and 16.
If you divide 16√2 in half you get 16√2/2, which is 1/2 x 16 x √2.
So 16√2/2 16√2/2 and 16 work as the sides and hypotenuse of an isosceles right triangle.
The perimeter of such a triangle is 16√2/2 + 16√2/2 + 16 = 16√2 + 16.
So the length of the hypotenuse of a right isosceles triangle the perimeter of which measures 16√2 + 16 is 16.
The efficiency of this method comes from knowing the ratios of the sides and hypotenuse of an isosceles right triangle and being able to recognize how to apply them.
Side Side Hypotenuse
..1.....1......√2
√2/2 √2/2...1
Looking at the second way, you can see that each side is 1/2 x length of the hypotenuse x √2.
Compare that relationship to the numbers given, 16√2 and 16.
If you divide 16√2 in half you get 16√2/2, which is 1/2 x 16 x √2.
So 16√2/2 16√2/2 and 16 work as the sides and hypotenuse of an isosceles right triangle.
The perimeter of such a triangle is 16√2/2 + 16√2/2 + 16 = 16√2 + 16.
So the length of the hypotenuse of a right isosceles triangle the perimeter of which measures 16√2 + 16 is 16.
The efficiency of this method comes from knowing the ratios of the sides and hypotenuse of an isosceles right triangle and being able to recognize how to apply them.
Marty Murray
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Contact me at [email protected] for a free consultation.
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.
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- Brent@GMATPrepNow
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An IMPORTANT point to remember is that, in any isosceles right triangle, the sides have length x, x, and x√2 for some positive value of x.The Perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle?
A) 8
B) 16
C) 4√2
D) 8√2
E) 16√2
Note: x√2 is the length of the hypotenuse, so our goal is to find the value of x√2
From here, we can see that the perimeter will be x + x + x√2
In the question, the perimeter is 16 + 16√2, so we can create the following equation:
x + x + x√2 = 16 + 16√2,
Simplify: 2x + x√2 = 16 + 16√2
IMPORTANT: Factor x√2 from the left side to get : x√2(√2 + 1) = 16 + 16√2
Now factor 16 from the right side to get: x√2(√2 + 1) = 16(1 + √2)
Divide both sides by (1 + √2) to get: x√2 = 16
Answer = B
Cheers,
Brent
- DavidG@VeritasPrep
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You can also backsolve:
If the hypotenuse were 16√2, the sides of the triangle would be 16, and the perimeter would be 16√2 + 16 + 16 = 16√2 + 32. That's too big.
If the hypotenuse were 8√2, the sides of the triangle would be 8, and the perimeter would be 8√2 + 8 + 8 = 8√2 + 16. That's too small.
So the answer has to be between 16√2 and 8√2. The only possibility is 16.
If the hypotenuse were 16√2, the sides of the triangle would be 16, and the perimeter would be 16√2 + 16 + 16 = 16√2 + 32. That's too big.
If the hypotenuse were 8√2, the sides of the triangle would be 8, and the perimeter would be 8√2 + 8 + 8 = 8√2 + 16. That's too small.
So the answer has to be between 16√2 and 8√2. The only possibility is 16.
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Hi Emeka N.,
Here's a larger discussion on this question, including other ways to solve it and some noteworthy aspects about how the prompt is written:
https://www.beatthegmat.com/hypotenuse-o ... 76892.html
GMAT assassins aren't born, they're made,
Rich
Here's a larger discussion on this question, including other ways to solve it and some noteworthy aspects about how the prompt is written:
https://www.beatthegmat.com/hypotenuse-o ... 76892.html
GMAT assassins aren't born, they're made,
Rich