Question

21 fill in the blank 1 point according to the diagram below, which similarity statements are true? choose your answer_ choose your answer_ triangle pqr - triangle prs triangle pqr - triangle qrs triangle pqs - triangle qrs previous

Explanation:

Step1: Recall the geometric mean theorem (or altitude-on-hypotenuse theorem)

In a right - triangle, if an altitude is drawn from the right - angle vertex to the hypotenuse, then the two smaller triangles formed are similar to the original triangle and to each other.

Step2: Analyze the given triangle

In \(\triangle PQS\) with right - angle at \(Q\) and altitude \(QR\) drawn to the hypotenuse \(PS\).
For \(\triangle PQR\) and \(\triangle QRS\):

• \(\angle PRQ=\angle QRS = 90^{\circ}\)
• \(\angle P+\angle PQR = 90^{\circ}\) and \(\angle PQR+\angle RQS=90^{\circ}\), so \(\angle P=\angle RQS\)

By the AA (angle - angle) similarity criterion (\(\angle PRQ=\angle QRS\) and \(\angle P=\angle RQS\)), \(\triangle PQR\sim\triangle QRS\)

Answer:

Triangle \(PQR\sim\) Triangle \(QRS\)