Solve Right Triangle Given Perimeter and Area - Problem With Solution

Solve a right triangle given its perimeter and area.

Problem:
ABC is a right triangle. The length of its perimeter is equal to 60 units and the size of its area is equal 150 units^{2}. Find its two sides and hypotenuse.

Solution to Problem :

The perimeter, the area and Pythagora theorems gives three equations as follows

a + b + h = 60

(1 / 2) a b = 150 or a b = 300

a^{ 2} + b^{ 2} = h^{ 2}

Rewrite the equation a + b + h = 60 as follows

a + b = 60 - h

Square both sides

(a + b)^{2} = (60 - h)^{2}

Expand both sides

a^{2} + b^{2} + 2 a b = 60^{2} + h^{2} - 120 h

Combine the equation a^{ 2} + b^{ 2} = h^{ 2} with the above equation to obtain

2 a b = 60^{2} - 120 h

a b is known to be equal to 300, hence the above equation becomes

600 = 60^{2} - 120 h

Solve for h to obtain

h = 25 units

Substitute h by 25 in the equation a + b + h = 60 to obtain

a + b = 60 - 25 = 35

Since a b = 300, then b = 300 / a which is substituted in the equation a + b = 35 to obtain

a + 300 / a - 35 = 0

Multiply all terms by a to obtain a quadratic equation of the form

a^{ 2} + 300 - 35 a = 0

Solve the above equation to obtain two solutions

a = 20 and a = 15

Use the equation a b = 300 to obtain

when a 20 , b = 15 and when a = 15 , b = 20

The two sides of the right triangle and the hypotenuse are, respectively, given by

15 units, 20 units and 25 units.

As an exercise check the perimeter given in the problem.