The leg of a right triangle is equal to 1/5 the sum of other sides. The triangle has a perimeter of 1. What is the triangle's area? Thank you.

Guest May 27, 2017

#1**+1 **

The leg of a right triangle is equal to 1/5 the sum of other sides. The triangle has a perimeter of 1. What is the triangle's area? Thank you.

Let the sides be a, b and c where a and b are the legs and c is the hypotenuse

Let a = ( b + c) / 5 so 5a = b + c

And since the perimeter = 1, we have that

a + b + c = 1

a + 5a = 1

6a = 1 → a = 1/6 units

And we have that 5(1/6)= b + c → 5/6 = b + c → c = (5/6) - b

So.....using the Pythagorean Theorem, we have that

a^2 + b^2 = c^2

(1/6)^2 + b^2 = [ (5/6) - b ] ^2

(1/36) + b^2 = b^2 - (10/6)b + 25/36 simplify

(1/36) = 25/36 - (5/3)b rearrange as

(5/3)b = 25/36 - 1/36

(5/3)b = 24/36

(5/3)b = 2/3

5b = 2

b = 2/5

So.......the area = (1/2)(product of the legs) = (1/2)(a)(b) = (1/2)(1/6)(2/5) = 1/30 units^2

CPhill
May 27, 2017

#2**0 **

let the base = b

let the hypotenuse = c

a=(b+c) / 5, or

5a =(b+c), but we have:.................(1)

a+b+c=1, we sub as

a+5a = 1, and we have:

a = 1/6, now will sub this into (1) above:

5/6 - b = c, Now to solve for b, will use Pytho. theorem:

a^2 + b^2 =c^2...........................(2)

(1/6)^2 + b^2 =(5/6 - b)^2 simplifying this we get:

(10/6)b = 24/36

b = 2/5, we now have:

Area = ab/2

**(1/6)(2/5)(1/2) =1/30 sq.units.**

Guest May 27, 2017

#3**+1 **

Another, albeit, intuitive way!

In a 5-12-13 triangle, 5=(12+13)/5. Its perimeter is 30, area =(5 x 12)/2 =30 sq.units.

The right triangle with a perimeter of 1 is similar to the 5-12-13 triangle. Its sides are then

1/30 of the 5-12-13, so its area is proportional by (1/30)^2. Its area is therefore =area(5-12-13) / 30^2

=30 /30^2 =1/30 sq.units.

Guest May 27, 2017

edited by
Guest
May 27, 2017