Question

ple 3 try it! use the converse of the isosceles triangle theorem

1. use the figure shown.

a. what is the value of x?

b. what are the lengths of all three sides of the triangle?

Explanation:

Step1: Apply the Converse of the Isosceles Triangle Theorem

Since the base angles are equal, the sides opposite them are equal. So, \(4x + 5=2x + 11\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(4x-2x + 5=2x-2x + 11\), which simplifies to \(2x+5 = 11\).
Subtract \(5\) from both sides: \(2x+5 - 5=11 - 5\), so \(2x=6\).
Divide both sides by \(2\): \(x=\frac{6}{2}=3\).

Step3: Find the lengths of the sides

For the side \(8x - 1\), substitute \(x = 3\): \(8\times3-1=24 - 1=23\).
For the side \(4x + 5\), substitute \(x = 3\): \(4\times3+5=12 + 5=17\).
For the side \(2x + 11\), substitute \(x = 3\): \(2\times3+11=6 + 11=17\).

Answer:

a. \(x = 3\)
b. The lengths of the sides are \(23\), \(17\), and \(17\).