Question

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

Explanation:

Step1: Recall Right Triangle Similarity

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 the altitude to hypotenuse \(SQ\). So, \(\triangle STR\), \(\triangle RTQ\), and \(\triangle SQR\) should be similar. Let's check the angle - angle (AA) similarity criterion.

Step2: Analyze Angles of \(\triangle STR\) and \(\triangle SQR\)

• For \(\triangle STR\) and \(\triangle SQR\):
• \(\angle S\) is common to both triangles.
• \(\angle STR=\angle SRQ = 90^{\circ}\) (since \(\triangle QRS\) is right - angled at \(R\) and \(RT\perp SQ\)).
• 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 \(\angle STR=\angle SRQ = 90^{\circ}\), \(\triangle STR\sim\triangle SQR\).

Now let's check the other options:

• Option 1: \(\triangle STR\) and \(\triangle TQR\): The angles do not match up correctly. \(\angle STR = 90^{\circ}\), \(\angle TQR\) is an acute angle in \(\triangle TQR\), and the common angle or the other angle does not satisfy AA similarity.
• Option 2: \(\triangle STR\) and \(\triangle RTQ\): The angle correspondence is not correct. The angles of \(\triangle STR\) and \(\triangle RTQ\) do not satisfy AA similarity.
• Option 3: \(\triangle STR\) and \(\triangle RST\): These are the same triangle (reflexive), not a non - trivial similarity statement for two different triangles as intended here.

Answer:

\(\boldsymbol{\triangle STR\sim\triangle SQR}\) (the fourth option: \(\triangle STR\sim\triangle SQR\))