Question

according to the diagram below, which similarity statements are true? choose your answer... choose your answer... triangle pqr ~ triangle qrs triangle pqr ~ triangle prs triangle pqs ~ triangle qrs

Explanation:

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

In a right - triangle \(PQS\) with altitude \(QR\) drawn to the hypotenuse \(PS\), we have the following similarity relationships.
For two right - triangles \(\triangle PQR\) and \(\triangle QRS\):

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

By the AA (angle - angle) similarity criterion, \(\triangle PQR\sim\triangle QRS\)

Step2: Analyze other options

• For \(\triangle PQR\) and \(\triangle PRS\): \(\triangle PRS\) is not a right - triangle (unless additional information is given which is not in the problem statement as per the basic geometric mean theorem set - up). The AA criterion for similarity is not met for \(\triangle PQR\) and \(\triangle PRS\) since \(\angle PRS

eq90^{\circ}\) (assuming standard right - triangle altitude - hypotenuse configuration where \(\angle PQS = 90^{\circ}\))

• For \(\triangle PQS\) and \(\triangle QRS\): \(\angle PQS = 90^{\circ}\) and \(\angle QRS=90^{\circ}\), but \(\angle S\) is common. However, \(\triangle PQS\) has angles \(\angle P\), \(\angle S\), \(90^{\circ}\) and \(\triangle QRS\) has angles \(\angle SQR\), \(\angle S\), \(90^{\circ}\). Since \(\angle P

eq\angle SQR\) (in the non - degenerate case of the right - triangle with altitude), \(\triangle PQS\) and \(\triangle QRS\) are not similar

Answer:

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