Question

△qrs is a right triangle. select the correct similarity statement. options: △str ~ △tqr, △str ~ △rst, △str ~ △sqr, △str ~ △rtq

Explanation:

Step1: Recall Similar Triangles in Right Triangles

In a right triangle, when an altitude is drawn to the hypotenuse, the two smaller triangles are similar to the original triangle and to each other. Here, $\triangle QRS$ is right - angled at $R$, and $RT$ is an altitude to hypotenuse $SQ$. So, $\triangle STR$, $\triangle RQT$, and $\triangle SQR$ (the original triangle) should be similar.

Step2: Analyze Angles for $\triangle STR$ and $\triangle SQR$

• For $\triangle STR$ and $\triangle SQR$:
• $\angle S$ is common to both $\triangle STR$ and $\triangle SQR$.
• $\angle STR=\angle SRQ = 90^{\circ}$ (since $RT\perp SQ$ and $\angle SRQ$ is the right angle of $\triangle QRS$).
• By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Since $\angle S=\angle S$ and $\angle STR=\angle SRQ = 90^{\circ}$, $\triangle STR\sim\triangle SQR$.
• Let's check other options:
• For $\triangle STR$ and $\triangle TQR$: The angles do not match up to satisfy AA similarity.
• For $\triangle STR$ and $\triangle RST$: These are the same triangle (improper similarity statement as they are not distinct in a way to show non - trivial similarity).
• For $\triangle STR$ and $\triangle RTQ$: The angle correspondence does not satisfy AA similarity.

Answer:

$\boldsymbol{\triangle STR\sim\triangle SQR}$