Question

1. an isosceles triangle has a base of 2 inches and sides of 3x and 2x+3. what is the perimeter of the triangle?

Explanation:

Step1: Use the property of isosceles triangle

Since it is an isosceles triangle, either \(3x = 2x + 3\) or \(3x=2\) or \(2x + 3=2\).
If \(3x=2\), then \(x=\frac{2}{3}\), and \(2x + 3=2\times\frac{2}{3}+ 3=\frac{4}{3}+3=\frac{13}{3}\). But in a triangle, sum of two sides must be greater than the third side. \(2 + 2=\ 4<\frac{13}{3}\approx4.33\) (not valid).
If \(2x + 3=2\), then \(2x=-1\), \(x =-\frac{1}{2}\) (side length cannot be negative, not valid).
If \(3x=2x + 3\), solve for \(x\).
\(3x-2x=2x + 3-2x\)
\(x = 3\)

Step2: Calculate the side lengths

When \(x = 3\), the sides are \(3x=3\times3 = 9\) and \(2x + 3=2\times3+3=9\)

Step3: Calculate the perimeter

Perimeter \(P=9 + 9+2\)
\(P=20\)

Answer:

\(20\) inches