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: Apply triangle inequality theorem

Let the length of the shortest straw be \(x\). Then \(12 - 9\lt x\lt12 + 9\), so \(3\lt x\lt21\). But since \(x\) is the shortest side, \(x\lt9\).

Step2: Use the obtuse - triangle condition

For a triangle with sides \(a=x\), \(b = 9\), \(c = 12\) (where \(c\) is the longest side), by the Pythagorean theorem for obtuse triangles \(a^{2}+b^{2}\lt c^{2}\). Substitute \(b = 9\) and \(c = 12\): \(x^{2}+9^{2}\lt12^{2}\), so \(x^{2}+81\lt144\), then \(x^{2}\lt144 - 81=63\), and \(x\lt\sqrt{63}\approx7.94\)

Step3: Combine the two conditions

We have \(3\lt x\lt7.94\) and \(x\lt9\).

Answer:

5 inches, 6 inches, 7 inches