Question

1. if \\( \triangle rst \\) is an equilateral triangle, find \\( x \\) and the measure of each side.\

\\( rs = 12x - 22 \\), \\( st = 10x - 6 \\), \\( rt = 7x + 18 \\)

Explanation:

Step1: Set two sides equal (equilateral triangle)

In an equilateral triangle, all sides are equal. So, set \(12x - 22 = 10x - 6\).
Subtract \(10x\) from both sides: \(12x - 10x - 22 = -6\) → \(2x - 22 = -6\).
Add 22 to both sides: \(2x = -6 + 22\) → \(2x = 16\).
Divide by 2: \(x = 8\).

Step2: Verify with another pair of sides

Now check with \(10x - 6\) and \(7x + 18\). Substitute \(x = 8\) into \(10x - 6\): \(10(8) - 6 = 80 - 6 = 74\).
Substitute \(x = 8\) into \(7x + 18\): \(7(8) + 18 = 56 + 18 = 74\). Also, check \(12x - 22\): \(12(8) - 22 = 96 - 22 = 74\). All sides equal 74.

Answer:

\(x = 8\), and each side measures \(74\).