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.
2. joanie is cutting pieces of colored string to create an obtuse triangle for an art project. the two shorter sides of the triangle measure 10 inches and 12 inches. can joanie create an obtuse triangle by using a 16 - inch piece of string for the third side? explain.

Explanation:

Step1: Determine the type of triangle for 14 - ft side

For a triangle with sides \(a = 8\), \(b=12\), \(c = 14\) (where \(c\) is the longest side).
Use the Pythagorean theorem \(a^{2}+b^{2}\) and \(c^{2}\)
\(a^{2}+b^{2}=8^{2}+12^{2}=64 + 144=208\)
\(c^{2}=14^{2}=196\)
Since \(a^{2}+b^{2}>c^{2}\) (i.e., \(208>196\)), the triangle is acute.

Step2: Determine the type of triangle for 15 - ft side

For a triangle with sides \(a = 8\), \(b = 12\), \(c=15\) (where \(c\) is the longest side)
\(a^{2}+b^{2}=8^{2}+12^{2}=64+144 = 208\)
\(c^{2}=15^{2}=225\)
Since \(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 (\(8^{2}+12^{2}=208<15^{2}=225\))