Question

identifying possible side lengths
marlena has 3 straws. two straws have the lengths shown. she does not know the length of the shortest straw, but when she forms a triangle with all three, the triangle is obtuse. which are possible lengths of the shortest straw? check all that apply.
5 inches
6 inches
7 inches
8 inches
9 inches

Explanation:

Step1: Set unknown side as $x$

Let the shortest straw length be $x$, so $x < 9$.

Step2: Triangle inequality check

For valid triangle: $x + 9 > 12 \implies x > 3$.

Step3: Obtuse triangle condition

Case 1: 12 is the longest side. For obtuse: $x^2 + 9^2 < 12^2$
$x^2 < 144 - 81 = 63 \implies x < \sqrt{63} \approx 7.94$
Case 2: 9 is longest (invalid, $x<9$ but $x+9>12$ gives $x>3$, but $9$ can't be longest obtuse here as $x^2 + 12^2 >9^2$ always true for positive $x$)

Step4: Narrow valid $x$

Combine $3 < x < 7.94$ and $x<9$. So $x=5,6$ fit.

Answer:

A. 5 inches, B. 6 inches