Question

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

Answer 1

Answer:10.2 inches

Explanation:

we know that

In this problem we have two cases

First case

The given lengths are two legs of the right triangle

so

Applying the Pythagoras Theorem

Find the length of the hypotenuse

substitute

Second case

The given lengths are one leg and the hypotenuse

so

Applying the Pythagoras Theorem

Find the length of the other leg

substitute

Find the difference between the two possible lengths of the third side of the triangle

so

Answer 2

The difference between the two possible lengths for the third side of the triangle is about 10.21 inches.

Explanation:

We are given that the lengths of two sides of a right triangle is 12 inches and 15 inches.

And we want to find the difference between the two possible lengths of the third side.

In the first case, assume that neither 12 nor 15 is the hypotenuse of the triangle. Then our third side c must follow the Pythagorean Theorem:

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

Substitute in known values:

`(12)^2+(15)^2=c^2`

Solve for c:

`c=√(12^2+15^2)=√(369)=√(9\cdot 41)=3√(41)`

In the second case, we will assume that one of the given lengths is the hypotenuse. Since the hypotenuse is always the longest side, the hypotenuse will be 15. Again, by the Pythagorean Theorem:

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

Substitute in known values:

`(12)^2+b^2=(15)^2`

Solve for b:

`b=√(15^2-12^2)=√(81)=9`

Therefore, the difference between the two possible lengths for the third side is:

`\displaystyle \text{Difference}=(3√(41))-(9)\approx 10.21\text{ inches}`