Question

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: Recall triangle - inequality theorem

For three side - lengths \(a\), \(b\), and \(c\) of a triangle, \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let the side - lengths of the triangle be \(a\), \(9\), and \(12\) (\(a\) is the shortest side, so \(a<9\)). Also, for an obtuse - angled triangle, if \(c\) is the longest side, then \(a^{2}+b^{2}12\) and \(a\) is large enough, \(a\) could be the longest side. But since \(a\) is the shortest side, the longest side is \(12\).

Step2: Apply the obtuse - triangle formula

We use the formula \(a^{2}+9^{2}<12^{2}\) (because for an obtuse triangle with sides \(a\), \(9\), and \(12\) where \(12\) is the longest side). So \(a^{2}+81 < 144\), then \(a^{2}<144 - 81=63\), and \(a < \sqrt{63}\approx 7.94\). Also, from the triangle - inequality \(a+9>12\), so \(a > 3\).

Step3: Check the options

For \(a = 5\): \(5^{2}+9^{2}=25 + 81 = 106<144\) and \(5+9>12\).
For \(a = 6\): \(6^{2}+9^{2}=36 + 81 = 117<144\) and \(6 + 9>12\).
For \(a = 7\): \(7^{2}+9^{2}=49+81 = 130<144\) and \(7 + 9>12\).
For \(a = 8\): \(8^{2}+9^{2}=64 + 81=145>144\), so it does not form an obtuse triangle.
For \(a = 9\): It is not the shortest side.

Answer:

5 inches, 6 inches, 7 inches