Question

write the coordinates of the vertices after a reflection over the line y = x.

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of \( F \), \( G \), and \( H \) from the graph.

• For \( F \): It's on the y - axis at \( (0, 6) \) (since x = 0, y = 6).
• For \( G \): It's on the y - axis at \( (0, 9) \) (wait, looking at the graph, the y - coordinate of G is 9? Wait, the grid: from the graph, G is at (0,9)? Wait, no, the original graph: the y - axis, F is at (0,6), G is at (0,9)? Wait, no, the user's graph: the y - axis, F is at (0,6), G is at (0,9)? Wait, no, the first grid line above F (which is at y = 6) is y = 7, 8, 9? Wait, the graph shows G at (0,9)? Wait, no, the original problem's graph: the y - axis, F is at (0,6), G is at (0,9)? Wait, no, looking at the coordinates: H is at (- 10,7)? Wait, no, the H is at (-10,7)? Wait, no, the user's graph: H is at (-10,7)? Wait, no, let's re - examine. The x - axis: H is at x=-10, y = 7? Wait, no, the first point: F is (0,6), G is (0,9), H is (-10,7). Wait, no, the y - coordinate of H: looking at the graph, H is at y = 7? Wait, the vertical grid lines: each square is 1 unit. So F: (0,6), G: (0,9), H: (-10,7).

Step2: Apply reflection over \( y = x \)

The rule for reflecting a point \( (x,y) \) over the line \( y=x \) is to swap the x and y coordinates, so the new point is \( (y,x) \).

• For point \( F(0,6) \):

After reflection over \( y = x \), we swap x and y. So \( F'=(6,0) \).

• For point \( G(0,9) \):

After reflection over \( y = x \), swap x and y. So \( G'=(9,0) \).

• For point \( H(-10,7) \):

After reflection over \( y = x \), swap x and y. So \( H'=(7,-10) \).

Answer:

\( F'(6, 0) \)
\( G'(9, 0) \)
\( H'(7, - 10) \)

Wait, wait, maybe I made a mistake in the original coordinates. Let's re - check the graph. Let's look at the y - axis: F is at (0,6), G is at (0,9)? Wait, no, the graph shows G at (0,9)? Wait, the user's graph: the y - axis, F is at (0,6), G is at (0,9), and H is at (-10,7). Wait, but maybe the original coordinates are: F(0,6), G(0,9), H(-10,7). Then reflection over y = x: (x,y)→(y,x). So F(0,6)→(6,0), G(0,9)→(9,0), H(-10,7)→(7,-10).

Alternatively, maybe H is at (-10,7)? Wait, the x - coordinate of H is - 10, y - coordinate is 7. Let's confirm the reflection rule: reflection over \( y=x \) maps \( (a,b) \) to \( (b,a) \). So if F is (0,6), then F' is (6,0). If G is (0,9), G' is (9,0). If H is (-10,7), H' is (7,-10).