Question

in the diagram below, rstu is a rectangle, and the two shaded regions are squares. if the length of sr is 4 m and the length of st is 8 m, what is the length of pq in meters? a √80 m b √24 m c √68 m d √144 m

Explanation:

Step1: Analyze the right - angled triangle

Since \(RSTU\) is a rectangle and the two shaded regions are squares, we can consider a right - angled triangle to find \(PQ\). The vertical side of the right - angled triangle formed with \(PQ\) as the hypotenuse has length \(8 - 4=4\) m and the horizontal side has length \(8 + 4 = 12\) m.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - angled triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(c=\sqrt{a^{2}+b^{2}}\). Here \(a = 4\) and \(b = 12\), so \(PQ=\sqrt{4^{2}+12^{2}}=\sqrt{16 + 144}=\sqrt{160}=\sqrt{16\times10}=4\sqrt{10}\approx\sqrt{16\times 5}= \sqrt{80}\) m.

Answer:

A. \(\sqrt{80}\) m