Right Triangle

Geometry Level 3

The length of the three sides of a right triangle are integers. One of its legs measures 17 cm. What is the perimeter of this triangle?


The answer is 306.

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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1 solution

Using the formulas to find a Pythagorean triple, we have the formulas for the legs a and b :

m 2 − n 2 = a m^2-n^2=a and 2 m n = b 2mn=b .

One leg measures 17. Since 17 is odd, therefore it must be the leg a :

m 2 − n 2 = 17 m^2-n^2=17 or ( m + n ) ( m − n ) = 17 (m+n)(m-n)=17 .

17 is prime then m + n = 17 m+n=17 and m − n = 1 m-n=1 . Now, we know that m = 9 m=9 and n = 8 n=8 .

Then the other leg has 2 ⋅ 9 ⋅ 8 = 144 2\cdot9\cdot8=144 cm, and the hypotenuse has m 2 + n 2 = 145 m^2+n^2=145 cm.

The perimeter of this triangle is 17 + 144 + 145 = 306 17+144+145=306 cm.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

There is a typo at the calculation of the hypotenuse. Correctly:

h = m 2 + n 2 = 9 2 + 8 2 = 81 + 64 = 145 h = m^2 + n^2 = 9^2 + 8^2 = 81 + 64 = 145

Also, the statement "How 17 is odd, then it must be the leg a:" should rather read "Since 17 is odd, therefore it must be the leg a:".

If the 17 would be just a side length (not necessarily a leg), then we would have another solution: the primitive Pythagorean triple (8, 15, 17) [m = 4, n=1] would give us the perimeter of 8 + 15 + 17 = 40.

Zee Ell - 4 years, 2 months ago

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Sorry about these mistakes, I need to practice more my English. I'll solve it on my pc

Victor Paes Plinio - 4 years, 2 months ago

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