Question

1. tony is putting a wooden border around the outside of his triangular vegetable garden. he wants to know if the garden is a right, acute, or obtuse triangle. two of the sides of the garden measure 8 feet and 12 feet in length. tony can choose a 14 - ft or a 15 - ft board for the third side. will both boards give him the same type of triangle? explain.

Explanation:

Step1: Recall the Pythagorean theorem and its converse

For a triangle with sides \(a\), \(b\), and \(c\) (\(c\) is the longest side), if \(a^{2}+b^{2}=c^{2}\), the triangle is right - angled; if \(a^{2}+b^{2}>c^{2}\), the triangle is acute - angled; if \(a^{2}+b^{2}

Step2: Check for \(c = 14\)

Calculate \(a^{2}+b^{2}=8^{2}+12^{2}=64 + 144=208\). Calculate \(c^{2}=14^{2}=196\). Since \(208>196\) (i.e., \(a^{2}+b^{2}>c^{2}\)), when \(c = 14\), the triangle is acute.

Step3: Check for \(c = 15\)

Calculate \(a^{2}+b^{2}=8^{2}+12^{2}=64 + 144 = 208\). Calculate \(c^{2}=15^{2}=225\). Since \(208<225\) (i.e., \(a^{2}+b^{2}

Answer:

No, the two boards will not give the same type of triangle. When the third side is \(14\) - ft, the triangle is acute (\(8^{2}+12^{2}=208>14^{2} = 196\)). When the third side is \(15\) - ft, the triangle is obtuse (\(8^{2}+12^{2}=208<15^{2}=225\)).