Question

1. reflect △hij across the line y = -x, then reflect △hij across the x - axis to create △hij. then rotate △hij 90° clockwise about the origin to create △hij.

h( , ) h( , ) h( , ) h( , )
i( , ) i( , ) i( , ) i( , )
j( , ) j( , ) j( , ) j( , )
coordinate connection
(x, y)→( , )→( , )→( , )

Explanation:

Step1: Reflect across $y = -x$

The rule for reflecting a point $(x,y)$ across the line $y=-x$ is $(x,y)\to(-y,-x)$.

Step2: Reflect across the x - axis

The rule for reflecting a point $(x,y)$ across the x - axis is $(x,y)\to(x, - y)$.

Step3: Rotate 90° clockwise about the origin

The rule for rotating a point $(x,y)$ 90° clockwise about the origin is $(x,y)\to(y,-x)$.

Let's assume the coordinates of $H=(x_1,y_1)$, $I=(x_2,y_2)$ and $J=(x_3,y_3)$.

For point $H$:

• After reflection across $y = -x$: $H'=(-y_1,-x_1)$
• After reflection across the x - axis: $H''=(-y_1,x_1)$
• After 90° clockwise rotation about the origin: $H'''=(x_1,y_1)$

For point $I$:

• After reflection across $y = -x$: $I'=(-y_2,-x_2)$
• After reflection across the x - axis: $I''=(-y_2,x_2)$
• After 90° clockwise rotation about the origin: $I'''=(x_2,y_2)$

For point $J$:

• After reflection across $y = -x$: $J'=(-y_3,-x_3)$
• After reflection across the x - axis: $J''=(-y_3,x_3)$
• After 90° clockwise rotation about the origin: $J'''=(x_3,y_3)$

Answer:

If $H=(x_1,y_1)$, $H'=(-y_1,-x_1)$, $H''=(-y_1,x_1)$, $H'''=(x_1,y_1)$
If $I=(x_2,y_2)$, $I'=(-y_2,-x_2)$, $I''=(-y_2,x_2)$, $I'''=(x_2,y_2)$
If $J=(x_3,y_3)$, $J'=(-y_3,-x_3)$, $J''=(-y_3,x_3)$, $J'''=(x_3,y_3)$
$(x,y)\to(-y,-x)\to(-y,x)\to(x,y)$