Question

a, c, b are vertices of a triangle, a, c, b are vertices of another triangle. options: a. rotation 90° clockwise about the origin; b. rotation 90° counterclockwise about the origin; c. reflection across the x - axis; d. reflection across the line y = x

Explanation:

Step1: Identify coordinates of original points

Let's find coordinates of \( A \), \( B \), \( C \). From the graph:

• \( A(-3, 1) \), \( B(-1, 2) \), \( C(-2, 1) \)
• After transformation, \( A'(-1, -3) \), \( B'(-2, -1) \), \( C'(-1, -2) \)

Step2: Test rotation rules

For a \( 90^\circ \) clockwise rotation about origin, the rule is \( (x, y) \to (y, -x) \).

• For \( A(-3, 1) \): \( (1, 3) \)? No, wait, wait. Wait, maybe I mixed up. Wait, \( 90^\circ \) clockwise: \( (x,y) \to (y, -x) \). Wait \( A(-3,1) \): \( y = 1 \), \( -x = 3 \)? No, that's not matching \( A'(-1, -3) \). Wait, maybe \( 90^\circ \) counterclockwise? Rule: \( (x,y) \to (-y, x) \).
• For \( A(-3,1) \): \( (-1, -3) \)? Wait \( -y = -1 \), \( x = -3 \)? No. Wait, maybe I made a mistake in coordinates. Wait, looking at the graph again. Wait, original triangle: \( A \) is at \( (-3,1) \), \( C \) at \( (-2,1) \), \( B \) at \( (-1,2) \). Transformed triangle: \( A' \) at \( (-1, -3) \), \( C' \) at \( (-1, -2) \), \( B' \) at \( (-2, -1) \). Wait, let's check rotation 90° clockwise: \( (x,y) \to (y, -x) \). So \( A(-3,1) \to (1, 3) \)? No. Wait 90° counterclockwise: \( (x,y) \to (-y, x) \). So \( A(-3,1) \to (-1, -3) \). Yes! Because \( -y = -1 \), \( x = -3 \)? Wait no, \( x=-3 \), \( y=1 \). So \( -y = -1 \), \( x = -3 \)? Wait, no, \( (-y, x) \) would be \( (-1, -3) \)? Wait \( x=-3 \), so \( (-y, x) = (-1, -3) \). Yes! That's \( A'(-1, -3) \). Let's check \( B(-1,2) \): \( (-2, -1) \). Using \( (-y, x) \): \( -y = -2 \), \( x = -1 \)? Wait no, \( x=-1 \), \( y=2 \). So \( -y = -2 \), \( x = -1 \)? Wait, no, \( (-y, x) \) is \( (-2, -1) \). Yes! \( B(-1,2) \to (-2, -1) \) which is \( B' \). \( C(-2,1) \): \( (-1, -2) \). \( -y = -1 \), \( x = -2 \)? Wait, \( (-y, x) = (-1, -2) \). Yes! So \( C(-2,1) \to (-1, -2) \) which is \( C' \). So the rotation is 90° counterclockwise about the origin. Let's check other options. Reflection across x-axis: \( (x,y) \to (x, -y) \). So \( A(-3,1) \to (-3, -1) \), not \( (-1, -3) \). Reflection across \( y=x \): \( (x,y) \to (y,x) \). \( A(-3,1) \to (1, -3) \), not matching. So the correct transformation is 90° counterclockwise about origin.

Answer:

B. rotation 90° counterclockwise about the origin