Question

1. find the area of the following triangle

Explanation:

Step1: Find the value of \(x\)

Since the triangle is isosceles (the two sides \(4x - 14\) and \(2x+6\) are equal in an isosceles triangle), we set up the equation \(4x - 14=2x + 6\).
Subtract \(2x\) from both sides: \(4x-2x-14=2x - 2x+6\), which simplifies to \(2x-14 = 6\).
Add \(14\) to both sides: \(2x-14 + 14=6 + 14\), so \(2x=20\).
Divide both sides by \(2\): \(x = 10\).

Step2: Calculate the lengths of the sides

Substitute \(x = 10\) into \(4x-14\): \(4\times10-14=40 - 14=26\).
Substitute \(x = 10\) into \(2x + 6\): \(2\times10+6=20 + 6=26\).

Step3: Use the area formula for a triangle \(A=\frac{1}{2}ab\sin C\)

Here \(a = 26\), \(b = 26\), and \(C = 30^{\circ}\), \(\sin30^{\circ}=\frac{1}{2}\).
So \(A=\frac{1}{2}\times26\times26\times\frac{1}{2}\).
First, \(26\times26 = 676\).
Then \(\frac{1}{2}\times676\times\frac{1}{2}=\frac{676}{4}=169\).

Answer:

The area of the triangle is \(169\).