Question

The lengths of two sides of a right triangle are 12 inches and 15 inches. What is the difference between the two possible lemgths of the third side of the triangle?

Answer 1

The difference between the two possible lengths of the third side of the triangle is 10.2 inches

Solution:

The lengths of two sides of a right triangle are 12 inches and 15 inches

To find: difference between the two possible lemgths of the third side of the triangle

Let us apply Pythagorean Theorem for right angled triangle

Pythagorean theorem, states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of other two sides of the right-angled triangle

Applying the Pythagorean Theorem for right angled triangle a, b and c

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

where

a and b are the legs of the triangle

c is the hypotenuse of the triangle

Here given that lengths of two sides of a right triangle are 12 inches and 15 inches

Case 1:

Assume that the third side is a leg, "b"

In this case we have

a = 12 inches

c = 15 inches

b = ?

Therefore, by pythogoras theorem,

`c^(2) =a^(2) +b^(2)\\\\Solve\ for\ b\\\\b^(2)=c^(2) -a^(2)\\\\\text{ substitute the values }\\\\b^(2)=15^(2)-12^(2)\\\\b^(2)=81\\\\b=9\ in`

Thus the length of third side is 9 inches

Case 2:

Assume that the third side is the hypotenuse

In this case we have

a = 12 inches

b = 15 inches

c = ?

substitute the values

`c^(2)=12^(2)+15^(2)\\\\\c^(2)=369\\\\c=19.2\ in`

Thus the length of third side is 19.2 inches

The difference between the two possible lengths of the third side of the triangle is:

difference = 19.2 inches - 9 inches

difference = 10.2 inches

Thus the difference between the two possible lengths of the third side of the triangle is 10.2 inches