Question

in the rectangle below, ( rv = 2x + 8 ), ( sv = 5x - 7 ), and ( mangle rvu = 68^{circ} ). find ( rt ) and ( mangle vst ).

Explanation:

Step1: Use the property of rectangle diagonals

In a rectangle, the diagonals are equal and bisect each other. So \(RV = SV\).

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Step2: Calculate \(RV\)

Substitute \(x = 5\) into \(RV=2x + 8\).
\(RV=2\times5+8=10 + 8=18\)
Since \(RT = 2RV\) (diagonals bisect each other), \(RT=2\times18 = 36\)

Step3: Find \(m\angle VST\)

In rectangle \(RUTS\), \(RS\parallel UT\), \(VS = VT\) (diagonals bisect each other). Also, \(\angle RVU\) and \(\angle SVT\) are vertical angles, so \(\angle SVT=\angle RVU = 68^{\circ}\).
Since \(VS = VT\), \(\triangle VST\) is isosceles. Let \(m\angle VST=m\angle VTS\). Using the angle - sum property of a triangle (\(m\angle VST+m\angle VTS+m\angle SVT = 180^{\circ}\)), and \(m\angle VST=m\angle VTS\)

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Answer:

\(RT = 36\)
\(m\angle VST=56^{\circ}\)