Question

applying triangle classification theorems
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 acute triangle rule

For a triangle with sides $a \leq b \leq c$, it is acute if $a^2 + b^2 > c^2$. Also, triangle inequality: $c < a + b$.

Step2: Define known and unknown sides

Let $a=3$, $b=7$, $c$ (longest side). First apply triangle inequality:
$c < 3 + 7 = 10$

Step3: Apply acute triangle condition

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

Step4: Ensure c is longest side

$c > 7$ (since 7 is the longer of the two existing sides)

Step5: Combine constraints

$7 < c < \sqrt{58}$

Answer:

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