Question

triangle ghj is graphed on a coordinate plane. the transformations listed below are performed on triangle ghj. • r: reflection across the line x = -1 • t: translation so that (x, y) → (x + 4, y + 3) select the letter that corresponds to the triangle shown on the coordinate plane below that is the result of performing transformation r, followed by transformation t, on triangle ghj. image of coordinate plane with triangles a, b, c, d a. triangle a b. triangle b c. triangle c d. triangle d

Explanation:

Step1: Identify coordinates of G, H, J

First, we find the coordinates of the original triangle \( GHJ \). From the graph:
- \( G(-9, 7) \)
- \( H(-5, 6) \)
- \( J(-8, 4) \)

Step2: Apply reflection \( R \) (across \( x = -1 \))

The formula for reflecting a point \( (x, y) \) across the line \( x = a \) is \( (2a - x, y) \). Here, \( a = -1 \), so the reflection of \( (x, y) \) is \( (2(-1) - x, y) = (-2 - x, y) \).

- For \( G(-9, 7) \): \( -2 - (-9) = 7 \), so \( G' = (7, 7) \)? Wait, no, wait. Wait, let's recalculate. Wait, \( x = -9 \), \( a = -1 \). The distance from \( x \) to \( a \) is \( |-9 - (-1)| = |-8| = 8 \). So the reflected \( x \)-coordinate is \( -1 + 8 = 7 \)? Wait, no, reflection over \( x = -1 \): the formula is \( x' = -1 + (-1 - x) = -2 - x \). Let's check with \( x = -9 \): \( -2 - (-9) = 7 \), yes. So \( G(-9,7) \) reflects to \( (7,7) \)? Wait, no, that seems off. Wait, maybe I made a mistake. Wait, let's take a point \( (x, y) \) and reflect over \( x = -1 \). The midpoint between \( x \) and \( x' \) is \( -1 \), so \( \frac{x + x'}{2} = -1 \implies x' = -2 - x \). So for \( G(-9,7) \): \( x' = -2 - (-9) = 7 \), \( y' = 7 \), so \( G' = (7,7) \). For \( H(-5,6) \): \( x' = -2 - (-5) = 3 \), \( y' = 6 \), so \( H' = (3,6) \). For \( J(-8,4) \): \( x' = -2 - (-8) = 6 \), \( y' = 4 \), so \( J' = (6,4) \). Wait, but looking at the graph, after reflection, maybe we need to check the next step.

Step3: Apply translation \( T \): \( (x, y) \to (x + 4, y + 3) \)

Now, apply the translation to the reflected points \( G' \), \( H' \), \( J' \).

- For \( G'(7,7) \): \( (7 + 4, 7 + 3) = (11, 10) \)
- For \( H'(3,6) \): \( (3 + 4, 6 + 3) = (7, 9) \)
- For \( J'(6,4) \): \( (6 + 4, 4 + 3) = (10, 7) \)

Wait, but looking at the graph, the triangles are labeled A, B, C, D. Let's check the coordinates of the triangles:

- Triangle A: Let's assume coordinates. Wait, maybe I made a mistake in the reflection. Wait, original \( G \) is at (-9,7), \( H \) at (-5,6), \( J \) at (-8,4). Let's re-express the reflection. The line \( x = -1 \) is a vertical line. The distance from \( G \) to \( x = -1 \) is \( |-9 - (-1)| = 8 \) units to the left. So reflection is 8 units to the right of \( x = -1 \), so \( x = -1 + 8 = 7 \), which matches. Then translation: \( x + 4 \), \( y + 3 \). So \( G' \) after reflection: (7,7), then translation: (11,10). \( H' \) after reflection: (3,6), translation: (7,9). \( J' \) after reflection: (6,4), translation: (10,7). Now, looking at the graph, triangle B has vertices around (11,10), (7,9), (10,7)? Wait, maybe the coordinates are different. Wait, maybe I misread the original coordinates. Let's check again.

Wait, original \( G \): looking at the grid, x=-9, y=7 (since x=-10 is left, so -9 is one right, y=7 is up 7). \( H \): x=-5, y=6. \( J \): x=-8, y=4. Then reflection over \( x = -1 \):

For \( G(-9,7) \): distance to \( x=-1 \) is \( |-9 - (-1)| = 8 \), so reflected x is \( -1 + 8 = 7 \), y=7: (7,7).

For \( H(-5,6) \): distance to \( x=-1 \) is \( |-5 - (-1)| = 4 \), so reflected x is \( -1 + 4 = 3 \), y=6: (3,6).

For \( J(-8,4) \): distance to \( x=-1 \) is \( |-8 - (-1)| = 7 \), so reflected x is \( -1 + 7 = 6 \), y=4: (6,4).

Then translation: (x+4, y+3):

\( G'' = (7+4, 7+3) = (11,10) \)

\( H'' = (3+4, 6+3) = (7,9) \)

\( J'' = (6+4, 4+3) = (10,7) \)

Now, looking at the graph, triangle B has vertices at (11,10), (7,9), (10,7)? Wait, maybe the labels are different. Alternatively, maybe I made a mistake in the reflection. Wait, maybe the original triangle is different. Wait, let's check the other way: maybe the reflection is done correctly, and then translation. Let's check the options.

Alternatively, maybe the original coordinates are:

Wait, maybe \( G \) is at (-9,7), \( H \) at (-5,6), \( J \) at (-8,4). After reflection over \( x=-1 \):

\( G' = (-2 - (-9), 7) = (7,7) \)

\( H' = (-2 - (-5), 6) = (3,6) \)

\( J' = (-2 - (-8), 4) = (6,4) \)

Then translation: (x+4, y+3):

\( G'' = (7+4,7+3) = (11,10) \)

\( H'' = (3+4,6+3) = (7,9) \)

\( J'' = (6+4,4+3) = (10,7) \)

Now, looking at the graph, triangle B is at the top right, with vertices around (11,10), (7,9), (10,7). So the answer should be B.

Answer:

B. triangle B