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?
options:

• greater than \\(\sqrt{58}\\) inches but less than 10 inches
• exactly \\(\sqrt{58}\\) inches
• not enough information given
• less than \\(\sqrt{58}\\) inches but greater than 7 inches

Explanation:

Step1: Recall acute triangle rule

For triangle with sides $a \leq b < c$, $a^2 + b^2 > c^2$

Step2: Set known sides

Let $a=3$, $b=7$, $c$ = longest side ($c>7$)

Step3: Apply acute condition

$3^2 + 7^2 > c^2$
$\implies 9 + 49 > c^2$
$\implies 58 > c^2$
$\implies c < \sqrt{58}$

Step4: Combine with triangle inequality

Since $c$ is longest, $7 < c < \sqrt{58}$

Answer:

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