Question

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

Explanation:

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

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

Step2: Apply the rotation rule

The rule for a \( 90^{\circ} \) counter - clockwise rotation about the origin is \( (x,y)\to(-y,x) \).
For point \( A(-4,1) \):
Substitute \( x=-4 \) and \( y = 1 \) into the rule \( (x,y)\to(-y,x) \), we get \( A'(-1,-4) \).
For point \( B(2,4) \):
Substitute \( x = 2 \) and \( y=4 \) into the rule \( (x,y)\to(-y,x) \), we get \( B'(-4,2) \).
For point \( C(3,1) \):
Substitute \( x = 3 \) and \( y = 1 \) into the rule \( (x,y)\to(-y,x) \), we get \( C'(-1,3) \).

Step3: Graph the original and rotated triangles

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

Answer:

The coordinates of \( \triangle ABC \) are \( A(-4,1) \), \( B(2,4) \), \( C(3,1) \) and the coordinates of its image after a \( 90^{\circ} \) counter - clockwise rotation about the origin are \( A'(-1,-4) \), \( B'(-4,2) \), \( C'(-1,3) \). Graph these points to obtain \( \triangle ABC \) and its rotated image.