Question

graph \\( \triangle a b c \\) and its image after a rotation \\( 90^{circ} \\) about the origin

Explanation:

Step1: Find coordinates of \( \triangle ABC \)

From the graph, \( A(-4,1) \), \( B(2,4) \), \( C(3,1) \).

Step2: Apply rotation rule

The rule for a \( 90^{\circ} \) rotation about the origin is \( (x,y)\to(-y,x) \).
For \( A(-4,1) \): \( x = - 4,y = 1 \), new coordinates \( A'(-1,-4) \).
For \( B(2,4) \): \( x = 2,y = 4 \), new coordinates \( B'(-4,2) \).
For \( C(3,1) \): \( x = 3,y = 1 \), new coordinates \( C'(-1,3) \).

Step3: Graph the triangles

Plot \( A(-4,1) \), \( B(2,4) \), \( C(3,1) \) for \( \triangle ABC \). Plot \( A'(-1,-4) \), \( B'(-4,2) \), \( C'(-1,3) \) for the rotated triangle.

Answer:

Graph \( \triangle ABC \) with vertices \( A(-4,1) \), \( B(2,4) \), \( C(3,1) \) and its image (rotated \( 90^{\circ} \) about the origin) with vertices \( A'(-1,-4) \), \( B'(-4,2) \), \( C'(-1,3) \).