Question

identifying possible side lengths
9 in.
12 in.
three, the triangle is obtuse. which are possible lengths of the shortest straw?
choose three correct answers.
5 inches
7 inches
6 inches
8 inches
9 inches

Explanation:

Step1: Find triangle inequality bound

Let the shortest side be $x$. Triangle inequality: $12 - 9 < x < 12 + 9$, so $3 < x < 21$. Also, $x \leq 9$ (since it's the shortest side).

Step2: Case1: 12 is the longest side (obtuse)

For obtuse triangle with longest side 12: $x^2 + 9^2 < 12^2$
$x^2 < 144 - 81 = 63$
$x < \sqrt{63} \approx 7.94$

Step3: Case2: x is not shortest, 9 is longest? No, x is shortest, so $x \leq 9$. If 9 were longest, $x^2 + 12^2 < 9^2$ which is impossible ($x^2 < -63$). So only case1 applies.

Step4: Combine bounds

$3 < x < 7.94$, and $x$ is positive integer. So $x=5,6,7$.

Answer:

A. 5 inches, C. 6 inches, B. 7 inches