Question

which set of transformations would generate the image of $\triangle abc$ with the coordinates $a(-2,-5), b(-2,-2), c(-6,-2)$?
options: a reflection over the $x$-axis followed by a translation 4 units to the left; a reflection over the $y$-axis followed by a translation 10 units down; a reflection over the $x$-axis followed by a rotation of $180^\circ$ about the origin; a reflection over the $y$-axis followed by a reflection over the $x$-axis

Explanation:

Step1: Find original coordinates

From the graph, original \( A(2,5) \), \( B(2,2) \), \( C(6,2) \).

Step2: Analyze each option

- **Option 1**: Reflect over \( x \)-axis: \( A(2,-5) \), \( B(2,-2) \), \( C(6,-2) \). Translate 4 left: \( A(-2,-5) \), \( B(-2,-2) \), \( C(2,-2) \) (not matching \( C'(-6,-2) \)).
- **Option 2**: Reflect over \( y \)-axis: \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \). Translate 10 down: \( A(-2,-5) \), \( B(-2,-2) \), \( C(-6,-2) \) (matches \( A', B', C' \)).
- **Option 3**: Reflect over \( x \)-axis: \( A(2,-5) \), \( B(2,-2) \), \( C(6,-2) \). Rotate \( 180^\circ \): \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \) (not matching).
- **Option 4**: Reflect over \( y \)-axis: \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \). Reflect over \( x \)-axis: \( A(-2,-5) \), \( B(-2,-2) \), \( C(-6,-2) \)? Wait, no—wait, original after \( y \)-axis reflection is \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \). Reflect over \( x \)-axis: \( A(-2,-5) \), \( B(-2,-2) \), \( C(-6,-2) \). Wait, but let's recheck Option 2: translate 10 down from \( y \)-axis reflection. \( y \)-axis reflection: \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \). Translate 10 down: subtract 10 from \( y \)-coordinate: \( 5 - 10 = -5 \), \( 2 - 10 = -8 \)? Wait, no! Wait, I made a mistake. Wait original \( B \) is \( (2,2) \), reflect over \( y \)-axis: \( (-2,2) \). Translate 10 down: \( y \)-coordinate \( 2 - 10 = -8 \), but \( B' \) is \( (-2,-2) \). Oh no, my mistake earlier. Wait let's re-express:

Wait original coordinates: \( A(2,5) \), \( B(2,2) \), \( C(6,2) \).

**Option 2: reflection over \( y \)-axis then translation 10 down.**

Reflection over \( y \)-axis: \( (x,y) \to (-x,y) \). So \( A(2,5) \to (-2,5) \), \( B(2,2) \to (-2,2) \), \( C(6,2) \to (-6,2) \).

Translation 10 down: \( (x,y) \to (x, y - 10) \). So \( A(-2,5 - 10) = (-2,-5) \), \( B(-2,2 - 10) = (-2,-8) \), but \( B' \) is \( (-2,-2) \). So that's wrong. Wait, so my earlier analysis was wrong.

Wait let's check Option 4: reflection over \( y \)-axis then reflection over \( x \)-axis.

Reflection over \( y \)-axis: \( A(-2,5) \), \( B(-2,2) \), \( C(-6,2) \).

Reflection over \( x \)-axis: \( (x,y) \to (x, -y) \). So \( A(-2,-5) \), \( B(-2,-2) \), \( C(-6,-2) \). Which matches \( A', B', C' \)! Wait, so I messed up Option 2. Let's recheck all:

- **Option 1**: Reflect over \( x \)-axis: \( (x,y) \to (x, -y) \). So \( A(2,-5) \), \( B(2,-2) \), \( C(6,-2) \). Translate 4 left: \( (x - 4, y) \). So \( A(2 - 4, -5) = (-2,-5) \), \( B(2 - 4, -2) = (-2,-2) \), \( C(6 - 4, -2) = (2,-2) \). \( C' \) is \( (-6,-2) \), so no.

- **Option 2**: Reflect over \( y \)-axis: \( (-2,5) \), \( (-2,2) \), \( (-6,2) \). Translate 10 down: \( (-2,5 - 10) = (-2,-5) \), \( (-2,2 - 10) = (-2,-8) \), \( (-6,2 - 10) = (-6,-8) \). Not matching \( B'(-2,-2) \), \( C'(-6,-2) \).

- **Option 3**: Reflect over \( x \)-axis: \( (2,-5) \), \( (2,-2) \), \( (6,-2) \). Rotate \( 180^\circ \): \( (-2,5) \), \( (-2,2) \), \( (-6,2) \). Not matching.

- **Option 4**: Reflect over \( y \)-axis: \( (-2,5) \), \( (-2,2) \), \( (-6,2) \). Reflect over \( x \)-axis: \( (-2,-5) \), \( (-2,-2) \), \( (-6,-2) \). Which matches \( A'(-2,-5) \), \( B'(-2,-2) \), \( C'(-6,-2) \). Wait, but earlier I thought Option 2 was wrong, and Option 4 is correct? Wait no, let's check the original graph again. Wait the original \( B \) is at \( (2,2) \), \( C \) at \( (6,2) \), \( A \) at \( (2,5) \). So after reflecting over \( y \)-axis: \( B(-2,2) \), \( C(-6,2) \), \( A(-2,5) \). Then reflecting over \( x \)-axis: \( B(-2,-2) \), \( C(-6,-2) \), \( A(-2,-5) \). Which is exactly \( A', B', C' \). So Option 4: "a reflection over the \( y \)-axis followed by a reflection over the \( x \)-axis" is correct? Wait but let's check the options again. Wait the options are:

1. reflection over \( x \)-axis followed by translation 4 left

2. reflection over \( y \)-axis followed by translation 10 down

3. reflection over \( x \)-axis followed by rotation \( 180^\circ \)

4. reflection over \( y \)-axis followed by reflection over \( x \)-axis

Wait, but when I did Option 4, it works. Wait but let's re-express the transformations:

- Reflect over \( y \)-axis: \( (x,y) \to (-x,y) \)

- Reflect over \( x \)-axis: \( (x,y) \to (x,-y) \)

Combined, this is equivalent to \( (x,y) \to (-x,-y) \), which is a \( 180^\circ \) rotation? Wait no, \( 180^\circ \) rotation is \( (x,y) \to (-x,-y) \), yes. But let's check the coordinates:

Original \( A(2,5) \to (-2,-5) \) (matches \( A' \))

Original \( B(2,2) \to (-2,-2) \) (matches \( B' \))

Original \( C(6,2) \to (-6,-2) \) (matches \( C' \))

Yes! So Option 4 is correct? Wait but earlier I thought Option 2 was wrong, which it is. So the correct option is the fourth one: "a reflection over the \( y \)-axis followed by a reflection over the \( x \)-axis".

Wait, but let's check the options again. The fourth option is: "a reflection over the \( y \)-axis followed by a reflection over the \( x \)-axis".

Yes, that works. Let's confirm each transformation:

1. Reflect \( \triangle ABC \) over \( y \)-axis:

- \( A(2,5) \) becomes \( A_1(-2,5) \)

- \( B(2,2) \) becomes \( B_1(-2,2) \)

- \( C(6,2) \) becomes \( C_1(-6,2) \)

2. Reflect \( \triangle A_1B_1C_1 \) over \( x \)-axis:

- \( A_1(-2,5) \) becomes \( A'(-2,-5) \)

- \( B_1(-2,2) \) becomes \( B'(-2,-2) \)

- \( C_1(-6,2) \) becomes \( C'(-6,-2) \)

Which matches the given \( A', B', C' \). So the correct option is the fourth one.

Answer:

The correct option is the fourth one: a reflection over the \( y \)-axis followed by a reflection over the \( x \)-axis (the last option in the list).