# Math

The side lengths of a right triangle are each an integral number of units. If one of the legs is 13 units, what is the perimeter of the triangle?

Thanks
~Zach

1. 👍 0
2. 👎 0
3. 👁 91
1. Look for three integers that satisfy
a^2 + b^2 = c^2

Have you tried 5, 12, 13? That is the only possibility with a 13-unit side.

Add them up for the perimeter.

1. 👍 0
2. 👎 0
posted by drwls
2. All Pythagorean Triples of the form x^2 + y^2 = z^2 derive from x = k(m^2 - n^2), y = k(2mn), and z = k(m^2 + n^2) where k is any positive integer and m and n are arbitrary positive integers, m greater than n. Pythagorean Triples that have no common factor, or a greatest common divisor of 1, are called primitive. Those with a common factor other than 1 are called non-primitive triples. Primitive Pythagorean Triples are obtained only when k = 1, m and n are of opposite parity (one odd one even) and have no common factor, and m is greater than n. (For x, y, & z to be a primitive solution, m and n cannot have common factors and cannot both be even or odd. Violation of these two limitations will produce non-primitive Pythagorean Triples.)

Assuming k = 1, and that your statement "If one of the legs is 13 units" means one of the legs at 90º to one another is 13 units, then 13 is the side which derives from m^2 - n^2. Quick observation shows that m must be 7 and n must be 6 making 13 = 7^2 - 6^2. With m = 7 and n = 6, the three sides become a = 7^2 - 6^2 = 13, b = 2(7)6 = 84 and c = 7^2 + 6^2 = 85.

Since the perfect squares are the sequential sum of the odd integers starting with 1, the sum of the 2nd and 3rd squares, 4 + 9 = 13 deriving from m = 3 and n = 2 which leads to another Pythagorean triangle where a = 3^2 - 2^2 = 5, b = 2(3)2 = 12 and c = 3^2 + 2^2 = 13.

No values of m an n can yield a side of 2mn as it always results in an even number.

1. 👍 0
2. 👎 0

## Similar Questions

1. ### math

an isosceles triangle has perimeter of 15m. Find all integral possibilities for the lengths of the side in meters. Hint: the sum of the lengths of any two sides of a triangle must exceed the third side.

asked by emily on October 17, 2018
2. ### algebra

Given a right triangle whose side lengths are integral multiples of 7, how many units are in the smallest possible perimeter

asked by algebra on January 20, 2011
3. ### maths

THE Perimeter of a triangle is 2004. One Side of a triangle = 21 Times the other. the shortest side is integral length. SOLVE For lengths of the sides of the traingle in every possible case.

asked by sonia on December 31, 2012
4. ### Math

An isosceles triangle has a perimeter of 15 m. Find all the integral possibilities for the lengths of the sides in meters. Hint the sum of the lengths of any two sides of a triangle must exceed the third side.

asked by emily on October 17, 2018
5. ### Math

If all of the following triangles have the same perimeter, which has the greater area? a) a right triangle with legs of equal lengths b) an equilateral triangle, c) An obtuse triangle d) a triangle whose sides are all different

asked by Anonymous on August 8, 2012
6. ### Pre-algebra (DESPRATE NEED OF HELP!)

A triangle has side lengths of (x+4), (4x-8), and (2x+8) units. If the perimeter of the triangle is at least 88 units, what is the minimum length of each side of the triangle?

asked by Ashley on May 23, 2011
7. ### Pre-algebra HELP ME PLEASE

A triangle has side lengths of (x+4), (4x-8), and (2x+8) units. If the perimeter of the triangle is at least 88units, what is the minimum length of each side of the triangle?

asked by Ashley on May 23, 2011
8. ### maths

1.how many non congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 times their perimeters? 2.a triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius

asked by ramesh reddy on May 7, 2013
9. ### Geometry

The sides of a triangle have lengths of 9, 11, and 16 units. What is the perimeter of a similar triangle with its longest side 24 units in length?

asked by Danielle on January 8, 2015
10. ### math

The lengths of the sides of a triangle are 13, 14 and 15 units. Find the shortest side of a similar triangle whose longest side is 21.

asked by Ana on July 9, 2015

More Similar Questions