Question

problem 1: isosceles triangle - park sign design
a city is designing a triangular park sign in the shape of an isosceles triangle. the two equal sides each measure 5 feet, and the base measures 6 feet. the city wants to add lights along all three sides of the sign.
question: how many feet of lights are needed in total for the park sign? also, is it possible to form this triangle with these side lengths?

Explanation:

Step1: Calculate the perimeter

The perimeter \(P\) of a triangle is the sum of its side lengths. For an isosceles triangle with two equal sides \(a = 5\) feet and base \(b=6\) feet, \(P=a + a+ b\).
Substitute \(a = 5\) and \(b = 6\) into the formula: \(P=5 + 5+6\).

Step2: Check the triangle inequality

The triangle - inequality theorem states that for any triangle with side lengths \(x\), \(y\), \(z\), the following must hold: \(x + y>z\), \(x+z > y\), and \(y + z>x\).
Let \(x = 5\), \(y = 5\), \(z = 6\).

• \(x + y=5 + 5=10>6\)
• \(x + z=5+6 = 11>5\)
• \(y + z=5 + 6=11>5\)

Answer:

The total length of lights needed (perimeter) is \(16\) feet. Yes, it is possible to form the triangle with side - lengths \(5\) feet, \(5\) feet, and \(6\) feet.