Question

Describe the possible side lengths of the 3rd side of the triangle given the lengths of the 2 other sides?

9 inches, 8 inches

Please explain! Thank you. :)

Answer 1

Heya !

Using a theoram about triangles ,

Given a triangle ∆ABC, the sum of the lengths of any two sides of the triangle is greater than the length of the third side ,

Also , the length of third side always greater than absolute difference of the other two sides ,

Let the third side be x ,

So , x < 9 + 8 and x > 9 - 8
x < 17 and x > 1

Hence , x ∈ [ 2 , 17 ] inch.

Above case is true for any triangle , be it scalene , Isosceles , Right-angled ...

As , for Isosceles , the third side can be 8 or 9 inches ,
For scalene , all values in the above range satusfies ,

For right angled triangle , we have 2 cases ,

Case 1 : Third side is the hypotenuse
Then , x = √(9²+8²) = √145 = 12.0415 inch.

Case 2 : Third side is not the hypotenuse
Then , x = √(9²-8²) = √17 = 4.1231 inch.

Hope it helps you ! :)

Answer 2

There are two possible choices for the third side. One is using the 9 as the hypotenuse, and the number less than 9. The second is using the number as they hypotenuse.
Remember to use the a² + b² = c² formula.

let the hypotenuse side be "x".

9² + 8² = x²
Simplify
81 + 64 = x²
Add
x² = 145
Root both sides
√x² = √145
Answer
x = 12.04159 (rounded)

let one of the sides (not hypotenuse) be x

9²- 8² = x²
Simplify
81 - 64 = x²
Subtract
x² = 17
Root both sides to isolate the x
√x² = √17
Answer
x = 4.12310 (rounded)

hope this helps