Question

identifying possible side lengths
marlená 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
9 in.
12 in.

Explanation:

Step1: Recall triangle inequality rule

For sides $a \leq b \leq c$, $a + b > c$; for obtuse triangle, $a^2 + b^2 < c^2$.

Step2: Case 1: 12 is the longest side

Let short side = $x$ ($x \leq 9$).
Triangle inequality: $x + 9 > 12 \implies x > 3$
Obtuse condition: $x^2 + 9^2 < 12^2$
$\implies x^2 < 144 - 81 = 63 \implies x < \sqrt{63} \approx 7.94$
So $3 < x < 7.94$, $x \leq 9$

Step3: Case 2: Unknown is longest side

Let short side = $x$, longest side = $x$ ($x \geq 9$).
Triangle inequality: $9 + 12 > x \implies x < 21$
Obtuse condition: $9^2 + 12^2 < x^2$
$\implies x^2 > 81 + 144 = 225 \implies x > 15$
But $x$ is the shortest side, so $x \leq 9$. This case is invalid.

Step4: Match valid values

Valid $x$: 5, 6, 7 (all $3 < x < 7.94$)

Answer:

□ 5 inches
□ 6 inches
□ 7 inches