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 and obtuse - triangle condition

For three side - lengths \(a\), \(b\), and \(c\) of a triangle (\(c\) is the longest side), the triangle - inequality theorem states that \(a + b>c\), \(a + c>b\), and \(b + c>a\). For an obtuse - triangle, \(a^{2}+b^{2}

Step2: Consider the cases when \(x\) is the shortest side

If \(x\) is the shortest side, and \(c = 12\), \(a=x\), \(b = 9\). From the triangle - inequality theorem, \(x+9>12\), so \(x > 3\). From the obtuse - triangle condition \(x^{2}+9^{2}<12^{2}\).
Solve the inequality \(x^{2}+81<144\), then \(x^{2}<144 - 81=63\), so \(x<\sqrt{63}\approx7.94\).
Combining with \(x>3\), the possible integer values of \(x\) are \(4,5,6,7\). Among the given options \(5\) inches, \(6\) inches, \(7\) inches satisfy the conditions.

Answer:

5 inches, 6 inches, 7 inches