Two of the sides of an acute triangle are shown. The length of the third side is also a positive integer. How many different possible values are there for the third side length? (Assume that the triangle is non-degenerate.)

(Two side lengths are 7 and 12)

surfz Aug 12, 2024

#1**+1 **

The problem states it must be an ACUTE triangle. The acute triangle states that the sum of the squares of two of the sides must always be greater than the square of the third side.

Essentially, it states that

\(7^2 +12^2 >x^ 2\\ 7^ 2 + x^ 2 > 1 2^ 2\\ \)

Now, solving each condition for x (gonna leave this up to you), we get that

\(x < \sqrt{193}\\ x > \sqrt{95}\)

Since x has to be an integer, we must round down for the first equation and round up for the second equation to get

\(x \leq 13\\ x \geq 10\)

Thus, we have \(10 \leq x \le 13\)

There are 4 numbers that work. 10, 11, 12,13.

I hope this explenation was clear enough to understand. Also, I'm not confident in my answer, but it should be right! :)

Thanks! :)

NotThatSmart Aug 12, 2024