Question

type the answers in the boxes below. 1. side pr measures units. 2. side sr measures units. 3. explain why triangle pqr is similar to triangle tsr. type your response in the space below.

Explanation:

Step1: Find side $PR$ using Pythagorean theorem in $\triangle PQR$

In right - triangle $PQR$ with $PQ = 40$ and $QR=75$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse. So $PR=\sqrt{40^{2}+75^{2}}=\sqrt{1600 + 5625}=\sqrt{7225}=85$.

Step2: Find side $SR$ using Pythagorean theorem in $\triangle TSR$

In right - triangle $TSR$ with $ST = 16$ and $RT = 34$, by the Pythagorean theorem $SR=\sqrt{34^{2}-16^{2}}=\sqrt{(34 + 16)(34 - 16)}=\sqrt{50\times18}=\sqrt{900}=30$.

Step3: Explain similarity of triangles

In $\triangle PQR$ and $\triangle TSR$, $\angle PQR=\angle TSR = 90^{\circ}$ (right - angles) and $\angle PRQ=\angle TRS$ (common angle). By the AA (angle - angle) similarity criterion, $\triangle PQR\sim\triangle TSR$.

Answer:

1. 85
2. 30
3. $\triangle PQR$ and $\triangle TSR$ are similar because $\angle PQR=\angle TSR = 90^{\circ}$ and $\angle PRQ=\angle TRS$ (AA similarity criterion).