Question

the given side lengths are: ( pq = x ), ( ps = 2x ), ( qr = 0.8x ), and ( rs = kx ), where ( k ) is a positive real number. what is the value of ( k ) rounded to the nearest thousandth? enter your answer in the box. your answer must be rounded to the nearest thousandth.

Explanation:

Step1: Find \(SQ\) using Pythagorean theorem in \(\triangle PQS\)

In right - triangle \(PQS\) with \(PQ = x\) and \(PS = 2x\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a = PQ\), \(b = PS\), and \(c = SQ\)), we have \(SQ=\sqrt{(2x)^{2}+x^{2}}=\sqrt{4x^{2}+x^{2}}=\sqrt{5x^{2}}=\sqrt{5}x\approx2.236x\).

Step2: Find \(RS\) using Pythagorean theorem in \(\triangle RQS\)

In right - triangle \(RQS\) with \(RQ = 0.8x\) and \(SQ=\sqrt{5}x\), by the Pythagorean theorem \(RS=\sqrt{(SQ)^{2}+(RQ)^{2}}\). Substitute \(SQ=\sqrt{5}x\) and \(RQ = 0.8x\) into the formula: \(RS=\sqrt{(\sqrt{5}x)^{2}+(0.8x)^{2}}=\sqrt{5x^{2}+0.64x^{2}}=\sqrt{5.64x^{2}}=\sqrt{5.64}x\). Since \(RS = kx\), then \(k=\sqrt{5.64}\).

Step3: Calculate the value of \(k\)

\(k=\sqrt{5.64}\approx2.375\)

Answer:

\(2.375\)