A special right triangle

Question. The length of one leg in a right triangle is equal to 1/5 the sum of the other two sides. The triangle has a perimeter of 1. What is the area of the triangle?

solution:


1st method:  
If you know some of the common right triangles (also known as primitive Pythagorean triples), then you might remember the 5-12-13 right triangle has the same property that one of its legs is 1/5 the sum of the other two: 5 = (12 + 13)/5.
You can guess, or prove, that the 5-12-13 triangle is similar to the triangle in the problem. As its perimeter is 30, the triangle with perimeter 1 has lengths that are 1/30 of the 5-12-13 triangle, so its area will be proportional by (1/30)² = 1/900.
As the 5-12-13 triangle has an area of (5)(12)/2 = 30, the triangle with perimeter 1 has an area of 30/900 = 1/30.

2nd method:


Answer To A Right Triangle With Perimeter 1
I came up with two ways to solve this problem. One method is to set up equations and work methodically.
Let the triangle have legs a and b and hypotenuse c. One leg is equal to 1/5 the sum of the other two sides, so we have:
a = (b + c)/5
5a = b + c
Then, because the perimeter is equal to 1, we have:
a + b + c = 1 a + (b + c) = 1 a + 5a = 1 a = 1/6
Substituting this value into the perimeter equation, we get:
a + b + c = 1 c = 1 – b – 1/6 c = 5/6 – b
Now we use the Pythagorean Theorem:
a² + b² = c²
1/36 + b² = (5/6 – b)²
1/36 + b² = 25/36 – (10/6) + b²
(10/6)b = 24/36
b = 2/5
The area of the triangle is then ab/2 = (1/6)(2/5)/2 = 1/30.