Question

figure efgh is rotated 90° clockwise around the origin to create figure pqrs.
image of coordinate grid with efgh
which of the following statements is true about this transformation.
○ the coordinates of the vertices of figure pqrs will be the same if figure efgh is rotated 270° counterclo
○ the rule for a 90° clockwise rotation is (x, y) → (-y, x).
○ angle g corresponds to angle s.
○ $overline{gh} = overline{sp}$

Explanation:

Step1: Analyze 90° clockwise and 270° counterclockwise rotations

A 90° clockwise rotation about the origin has the rule \((x,y)\to(y, -x)\). A 270° counterclockwise rotation about the origin is equivalent to a 90° clockwise rotation (since \(360 - 270=90\) and counterclockwise 270° is same as clockwise 90° in terms of final position). Wait, actually, the rule for 270° counterclockwise is \((x,y)\to(y, -x)\) (same as 90° clockwise). Wait, no: Let's recall the rotation rules.

• 90° clockwise: \((x,y)\to(y, -x)\)
• 90° counterclockwise: \((x,y)\to(-y, x)\)
• 180° rotation: \((x,y)\to(-x, -y)\)
• 270° clockwise (or 90° counterclockwise): \((x,y)\to(-y, x)\)
• 270° counterclockwise (or 90° clockwise): \((x,y)\to(y, -x)\)

Wait, correction: 270° counterclockwise rotation about the origin is the same as 90° clockwise rotation? No, wait: The angle of rotation: a full rotation is 360°. So 270° counterclockwise is equal to \(360 - 270 = 90\)° clockwise? No, that's not right. Let's think in terms of direction. Counterclockwise 270° is the same as clockwise 90°? Wait, no. Let's take a point \((1,0)\). Rotate 90° clockwise: \((0, -1)\). Rotate 270° counterclockwise: Let's see, 90° counterclockwise: \((0,1)\), 180° counterclockwise: \((-1,0)\), 270° counterclockwise: \((0, -1)\). Oh! So 270° counterclockwise rotation is the same as 90° clockwise rotation. So the first option: "The coordinates of the vertices of figure \(PQRS\) will be the same if figure \(EFGH\) is rotated \(270^\circ\) counterclockwise" – since 90° clockwise and 270° counterclockwise give the same result, this statement is true? Wait, let's check other options.

Step2: Analyze the rotation rule option

The rule for 90° clockwise rotation is \((x,y)\to(y, -x)\), not \((x,y)\to(-y, x)\) (that's 90° counterclockwise or 270° clockwise). So the second option is false.

Step3: Analyze corresponding angles

When a figure is rotated, the corresponding angles: \(EFGH\) rotated 90° clockwise to \(PQRS\). So vertex \(E\) corresponds to \(P\), \(F\) to \(Q\), \(G\) to \(R\), \(H\) to \(S\). So angle \(G\) corresponds to angle \(R\), not \(S\). So third option is false.

Step4: Analyze segment lengths

Rotation is a rigid transformation, so lengths of segments are preserved. \(\overline{GH}\) should correspond to \(\overline{RS}\), not \(\overline{SP}\). So fourth option is false.

So the first option is true.

Answer:

The coordinates of the vertices of figure \(PQRS\) will be the same if figure \(EFGH\) is rotated \(270^\circ\) counterclockwise (the first option)