Question

ramon wants to make an acute triangle with three pieces of wood. so far, he has cut wood lengths of 7 inches and 3 inches. he still needs to cut the longest side. what length must the longest side be in order for the triangle to be acute?
exactly \\(\sqrt{58}\\) inches
greater than \\(\sqrt{58}\\) inches but less than 10 inches
less than \\(\sqrt{58}\\) inches but greater than 7 inches
not enough information given

Explanation:

Step1: Recall triangle inequality

For sides $a,b,c$ (longest $c$): $a+b>c$
Here $a=3, b=7$, so $3+7>c \implies c<10$

Step2: Recall acute triangle rule

For acute triangle (longest $c$): $a^2+b^2>c^2$
Substitute values: $3^2+7^2>c^2$
$\implies 9+49>c^2 \implies 58>c^2 \implies c<\sqrt{58}$

Step3: Confirm longest side condition

Since $c$ is the longest side, $c>7$ (as 7 > 3)

Step4: Combine conditions

$7 < c < \sqrt{58}$

Answer:

less than $\sqrt{58}$ inches but greater than 7 inches