Track / Stage 3 / 31 of 260 #31 of 2000

Problem 31

AMC 10/12, early questions

Geometry Difficulty 3.1 Find the answer

For how many integers $x$ does a triangle with side lengths $10, 24$ and $x$ have all its angles acute?
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ \text{more than } 7$

Your answer

Official solution

We first notice that the sides $10$ and $24$ , can be part of $2$ different right triangles, one with sides $10,24,26$ , and the other with a leg
somewhere between $21$ and $22$ . We now notice that if $x$ is less than or equal to $21$ , one of the angles is obtuse, and that the same is the same for
any value of $x$ above $26$ . Thus the only integer values of $x$ that fit the conditions, are $x=22, 23, 24, \text{and }25.$
So, the answer is $\boxed{\text{A}}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.