Question

1. aqrs is an equilateral triangle. if qr is seven less than twice x, rs is 61 less than five times x, and qs is 11 more than x, find the value of x and the measure of each side.

Explanation:

Step1: Set up equations based on the properties of an equilateral triangle

Since \( \triangle QRS\) is equilateral, \(QR = RS=QS\).
Given \(QR = 2x - 7\), \(RS = 5x-61\), and \(QS=x + 11\).
Set \(QR = RS\):
$$2x-7=5x - 61$$

Step2: Solve the equation \(2x-7=5x - 61\) for \(x\)

Subtract \(2x\) from both sides:
$$-7 = 3x-61$$
Add \(61\) to both sides:
$$54=3x$$
Divide both sides by \(3\):
$$x = 18$$

Step3: Find the length of each side

Substitute \(x = 18\) into \(QR=2x - 7\):
$$QR=2\times18-7=36 - 7=29$$
Substitute \(x = 18\) into \(RS = 5x-61\):
$$RS=5\times18-61=90 - 61=29$$
Substitute \(x = 18\) into \(QS=x + 11\):
$$QS=18 + 11=29$$

Answer:

\(x = 18\), and the length of each side (\(QR\), \(RS\), \(QS\)) is \(29\)