Question

Two sides of a triangle have lengths 10 and 15. What must be true about the length of the third side?

Answer 1

The length of the third side must be less than 25 units.

Step-by-step explanation:

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Given that two sides of the triangle have lengths 10 and 15, we can determine the possible range for the length of the third side.

Let x represent the length of the third side. According to the triangle inequality theorem, we have the following inequality:

10 + 15 > x

25 > x

Therefore, the length of the third side must be less than 25 units.

Answer 2

A widely-known theorem can be used for this, but only if it is a right triangle (let's assume it is).

Pythagorean Theorem, or a​^2+b^2=c^2 (the ^ means it is brought up to that power, so it would be a(squared) + b(squared) = c(squared))

Let's plug in our numbers into this equation to see what we get,

(Assuming that 10 and 15 are the legs of the triangle.)
10^2 + 15^2 = c^2 (c is your hypotenuse, or your longest side of the triangle.)

10^2 = 100 & 15^2 = 225
100 + 225 = c^2
325 = c^2
(square root)325 = c
18.03 ≈ c

So, your final equation should look like this:

10^2 + 15^2 = 18.03^2
Which it does! 100 + 225 = 325

What must be true about the length of the third side?
The third side must equal 18.03, only if it is a right triangle.