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.
9 in.
12 in.
5 inches
6 inches
7 inches
8 inches
9 inches

Explanation:

Step1: Set unknown side as $x$

Let $x$ = shortest straw length, so $x < 9$.

Step2: Triangle inequality rule

Sum of two sides > third side:
$x + 9 > 12$
$x > 12 - 9$
$x > 3$

Step3: Obtuse triangle condition

For obtuse triangle, either:
Case1: Longest side is 12:
$x^2 + 9^2 < 12^2$
$x^2 < 144 - 81$
$x^2 < 63$
$x < \sqrt{63} \approx 7.94$
Case2: Longest side is 9:
$x^2 + 12^2 < 9^2$
$x^2 < 81 - 144$ (impossible, negative)

Step4: Combine valid range

$3 < x < 7.94$, so $x=5,6$.

Answer:

5 inches, 6 inches