Question

in the diagram below of triangle qrs, t is the mid - point of $overline{qs}$ and u is the mid - point of $overline{rs}$. if m$angle sqr=100 - 2x$, and m$angle stu = 8x + 50$, what is the measure of $angle sqr$?

Explanation:

Step1: Apply mid - point theorem

Since \(T\) is the mid - point of \(\overline{QS}\) and \(U\) is the mid - point of \(\overline{RS}\), by the mid - point theorem of a triangle, \(TU\parallel QR\). When two parallel lines are cut by a transversal (\(QS\) in this case), corresponding angles are equal. So, \(\angle SQR=\angle STU\).

Step2: Set up the equation

Set \(100 - 2x=8x + 50\).

Step3: Solve the equation for \(x\)

Add \(2x\) to both sides: \(100=8x + 2x+50\), which simplifies to \(100 = 10x+50\). Then subtract 50 from both sides: \(100 - 50=10x\), so \(50 = 10x\). Divide both sides by 10: \(x = 5\).

Step4: Find the measure of \(\angle SQR\)

Substitute \(x = 5\) into the expression for \(\angle SQR\): \(m\angle SQR=100-2x=100 - 2\times5=100 - 10 = 90\).

Answer:

90