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?
problem 2: triangle inequality - bridge supports
an engineer is planning to build a triangular support structure under a bridge. the lengths of two sides of the triangle are 10 meters and 7 meters. the third side has to be chosen carefully to make a valid triangle.
question: what is the range of possible lengths for the third side so that the structure forms a valid triangle?

Explanation:

Problem 1:

Step1: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=a + b+ c\). Here \(a = 5\), \(b = 5\), \(c=6\).
\(P=5 + 5+6\)

Step2: Check the triangle - inequality

The triangle - inequality states that for a triangle with side lengths \(x\), \(y\), \(z\): \(x + y>z\), \(x+z > y\), \(y + z>x\)

• \(5+5>6\) (i.e., \(10>6\))
• \(5 + 6>5\) (i.e., \(11>5\))
• \(5+6>5\) (i.e., \(11>5\))

Step1: Apply the triangle - inequality theorem

Let the side lengths be \(a = 10\), \(b = 7\), and \(c\) be the third side.
By the triangle - inequality \(a + b>c\) and \(|a - b|First, \(a + b>c\) gives \(10+7>c\) (i.e., \(c < 17\))
Second, \(|a - b|

Answer:

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

Problem 2: