Question

no=61, op=55, np=42, rs=33, and qs=25.2, find the perimeter of △qrs. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since two angles of \(\triangle NOP\) are equal to two angles of \(\triangle QRS\) (by angle - angle similarity criterion), \(\triangle NOP\sim\triangle QRS\).

Step2: Find the scale factor

The scale factor \(k\) is given by \(\frac{RS}{OP}\). Given \(RS = 33\) and \(OP=55\), so \(k=\frac{33}{55}=\frac{3}{5}\).

Step3: Find the length of \(QR\)

We know that \(\frac{QR}{NP}=\frac{3}{5}\). Given \(NP = 42\), then \(QR=\frac{3}{5}\times42 = 25.2\).

Step4: Find the length of \(QS\)

We know that \(\frac{QS}{NO}=\frac{3}{5}\). Let \(NO=x\). But we can also use the perimeter formula. The perimeter of \(\triangle NOP\) is \(NO + OP+NP\). Let's first find \(NO\) using the Law of Sines (but since we have similarity, another approach). The perimeter of \(\triangle NOP\) is \(NO + 55+42\). Let's use the ratio.
The perimeter of \(\triangle NOP\): \(NO + 42+55\). Let's find \(NO\) from the angle - angle similarity. But another way:
The perimeter of \(\triangle NOP\): \(P_{NOP}=NO + 42 + 55\). Using the ratio of sides (similarity ratio \(k=\frac{3}{5}\)).
The perimeter of \(\triangle QRS\): \(P_{QRS}=QR + RS+QS\).
Since \(\frac{P_{QRS}}{P_{NOP}}=\frac{3}{5}\).
First, find \(NO\) from \(\frac{QS}{NO}=\frac{3}{5}\), given \(QS = 25.2\), then \(NO=\frac{25.2\times5}{3}=42\).
The perimeter of \(\triangle NOP\) is \(42+42 + 55=139\).
Then \(P_{QRS}=\frac{3}{5}\times139=83.4\).

Answer:

The perimeter of \(\triangle QRS\) is \(83.4\)