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How many right triangles are there using 3 lines among 7 lines with lengths of 5, 6, 7, 8, 10, 12, and 13?

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by Max@Math Revolution » Thu Mar 26, 2020 3:52 am

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[GMAT math practice question]

How many right triangles are there using 3 lines among 7 lines with lengths of 5, 6, 7, 8, 10, 12, and 13?

A. 1
B. 2
C. 3
D. 4
E. 5
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Re: How many right triangles are there using 3 lines among 7 lines with lengths of 5, 6, 7, 8, 10, 12, and 13?

by deloitte247 » Sun Mar 29, 2020 2:07 pm
Given there are 7 lines with lengths 5,6,7,8,10, 12 and 13
Find the right angled triangle that can be found using three lines among the given 7 lines
All right angle Triangles follow pythagoras triplet which is a set of positive integers a,b and c that satisfies and fit the rule
$$a^2+b^2=c^2$$ where a, b and c are the Length.

Therefore, we are looking at the pythagoras triplet that fall in the in the in the range of the given lengths.
Starting from the smallest pythagoras theorem, which is 3, 4 and 5, Length 3 and 4 are not among the given the length but if we scale it up by 2 i.e (3*2), (4*2) and (5*2)= 6,8 and 10 this is also a triple third can form a right angle triangle since all the length are among the three given length.

All the pythagoras triplets (scale and unscaled) have lengths that cannot be obtained from the given set of the lengths.
All the 7 Lengths consists of different pythagoras triplets that can be used to form 2 right angled
triangles

$$Answer\ is\ OptionB$$
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Re: How many right triangles are there using 3 lines among 7 lines with lengths of 5, 6, 7, 8, 10, 12, and 13?

by Max@Math Revolution » Sun Mar 29, 2020 9:36 pm
=>

Assume c is the length of the hypotenuse of the right triangle, while a and b are the lengths of the legs with a < b.
If c = 13, then a = 5, b =1 2 is the only possible solution.
If c = 12, then there is no possible solution of a and b.
If c = 10, then a = 6, b = 8 is the only possible solution.
If c = 8, then there is no possible solution of a and b.

Therefore, B is the answer.
Answer: B
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