Question

7 given \\( \triangle a b c \\) with points \\( a(4,3), b(4,-2) \\), and \\( c(2,3) \\). on the grid below, sketch \\( \triangle a b c \\). on the same set of axes, graph and state the coordinates of \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\), the image of \\( \triangle a b c \\) after a reflection in the line \\( y = x \\). on the same set of axes, graph and state the coordinates of \\( \triangle a ^ { \prime \prime } b ^ { \prime \prime } c ^ { \prime \prime } \\), the image of \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\) after the translation \\( t _ { - 4,3 } \\).

Explanation:

Step1: Reflect over \( y = x \)

To reflect a point \((x, y)\) over the line \( y = x \), we swap the \( x \)- and \( y \)-coordinates. So for each point of \( \triangle ABC \):

• For \( A(4, 3) \), the reflection \( A' \) is \( (3, 4) \).
• For \( B(4, -2) \), the reflection \( B' \) is \( (-2, 4) \).
• For \( C(2, 3) \), the reflection \( C' \) is \( (3, 2) \).

Step2: Translate \( T_{-4, 3} \)

A translation \( T_{h, k} \) means we add \( h \) to the \( x \)-coordinate and \( k \) to the \( y \)-coordinate of each point. Here, \( h = -4 \) and \( k = 3 \).

• For \( A'(3, 4) \), the translated point \( A'' \) is \( (3 + (-4), 4 + 3) = (-1, 7) \).
• For \( B'(-2, 4) \), the translated point \( B'' \) is \( (-2 + (-4), 4 + 3) = (-6, 7) \).
• For \( C'(3, 2) \), the translated point \( C'' \) is \( (3 + (-4), 2 + 3) = (-1, 5) \).

Answer:

• Coordinates of \( \triangle A'B'C' \): \( A'(3, 4) \), \( B'(-2, 4) \), \( C'(3, 2) \)
• Coordinates of \( \triangle A''B''C'' \): \( A''(-1, 7) \), \( B''(-6, 7) \), \( C''(-1, 5) \)