# Two sides of a right triangle are equal to 8 and 10. What is the third side? How many results are possible?

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Two sides of a right triangle are equal to 8 and 10. The third side has to be determined. This side can either be the shortest side, or one that lies between 8 and 10 or the longest side. For each of the three cases, if the length of the unknown side is X, we have:

1. `X^2 + 8^2 = 10^2`

=> `X^2 = 100 - 64`

=> `X^2 = 36`

=> X = 6

X can only take on the value 6 that satisfies the Pythagorean theorem while X is not the longest side.

2. X^2 = 8^2 + 10^2

=> X^2 = 64 + 100

=> X^2 = 164

=> X = `sqrt 164`

**There are two values of the length of the third side, 6 and `sqrt 164` .**

Rigtht angle triangles can be constructed with 2 configurations using line segments of length 8 and 10.

**Case 1**. *When one side = 8,hypotenuse = 10 and 2nd side = x*

8^2 + x^2 = 10^2

x^2 = 100-64 = 36

**x = 6**

**Case 2.** *When one side 8, other side = 10 and hypotenuse = x*

using paythogoras theorem:

x^2 = 8^2 + 10^2

x^2 = 164

**x = square root of 164**

*A right angle triangle cannot exist with line segment of length 8 as hypotenuse as it is shorter than the line the side of length 10.*

You have to use paythagoras therum to find the answer.

As 8 is a lower value than 10, it will be the short side, and ten will by the long side (hyptenuse)

so you do x to the power of 2 + 8 to the power of 2 will equal 10 to the power of two. Therefore to find the value of x you do square root of 10 to the power of two minus 8 to the power of 2. Which will give you 6. or you could write it as the square root of 164.

Hope this helps