Question

How do you find the perimeter of a right triangle with the area 9 inches squared?

Answer 1

Hint: When two sides of a triangle are perpendicular to each other, the triangle is known as a right-angled triangle or simply a right triangle. According to the Pythagoras theorem, the sum of the squares of the base and the perpendicular is equal to the square of the hypotenuse that’s why it is greater than both the base and the perpendicular. Thus by Pythagoras theorem, we can find out the value of the hypotenuse, after finding out the length of the base and the height from the area of the triangle.

Complete step-by-step answer:
As perimeter is the sum of the length of the three sides of a triangle, we can find it easily by finding out the sides
Area of a right-angled triangle,
 $
  Area = \dfrac{1}{2} \times base \times height \\
   \Rightarrow \dfrac{1}{2} \times b \times h = 9 \\
   \Rightarrow b \times h = 18 \;
  $
Now, 18 is the product of 18 and 1; 9 and 2; and 6 and 3, then b and h = 1 and 18, 2 and 9, or 3 and 6.
When b and h is equal to 1 and 18 –
 $
  hypotenuse = \sqrt {{1^2} + {{18}^2}} = 5\sqrt {13} \approx 18 \\
   \Rightarrow Perimeter = 1 + 18 + 15 = 34\,inches \;
  $
When b and h is equal to 9 and 2 –
 $
  hypotenuse = \sqrt {{9^2} + {2^2}} = \sqrt {85} \approx 9.2 \\
   \Rightarrow Perimeter = 9 + 2 + 9.2 = 20.2\,inches \;
  $
When b and h is equal to 3 and 6 –
 $
  hypotenuse = \sqrt {{3^2} + {6^2}} = 3\sqrt 5 \approx 6.7 \\
   \Rightarrow Perimeter = 3 + 6 + 6.7 = 15.7\,inches \;
  $
Hence, the perimeter of a right triangle with the area 9 inches squared is 34, 20.2, or 15.7 inches.
So, the correct answer is “34, 20.2, or 15.7 inches.”.

Note: We can’t find the value of two unknown quantities using a single equation, so we find out the pair of numbers whose product is 18, the pair of numbers will give us values that the base and the height of the right-angled triangle can have. By putting the different values of base and height, we get different values of the hypotenuse, so, we get different values of the perimeter of the given triangle.