Question

Pentagon ABCDE and pentagon A″B″C″D″E″ are shown on the coordinate plane below:

Pentagon ABCDE and pentagon A double prime B double prime C double prime D double prime E double prime on the coordinate plane with ordered pairs at A negative 5, negative 2, at B negative 7, negative 3, at C negative 6, negative 6, at D negative 3, negative 5, at E negative 3, negative 3, at A prime 3, 6, at B prime 5, 5, at C prime 4, 2, at D prime 1, 3, at E prime 1, 5.

Which two transformations are applied to pentagon ABCDE to create A″B″C″D″E″?

Translated according to the rule (x, y) → (x + 8, y + 2) and reflected across the y-axis
Translated according to the rule (x, y) → (x + 2, y + 8) and reflected across the x‒axis
Translated according to the rule (x, y) → (x + 2, y + 8) and reflected across the y-axis
Translated according to the rule (x, y) → (x + 8, y + 2) and reflected across the x‒axis

Answer 1

To identify the transformations applied to pentagon ABCDE to create A″B″C″D″E″, let's examine the coordinates of points A, B, C, D, and E, and the corresponding transformed points A″, B″, C″, D″, and E″.

Original points:
- A(-5, -2)
- B(-7, -3)
- C(-6, -6)
- D(-3, -5)
- E(-3, -3)

Transformed points:
- A″(3, 6)
- B″(5, 5)
- C″(4, 2)
- D″(1, 3)
- E″(1, 5)

### Step 1: Determine Translation
To find the translation, we compare a corresponding point in the original pentagon and the transformed pentagon.

Taking point A to point A″:
- A(-5, -2) to A″(3, 6)

Calculating the change in coordinates:
- Change in x: 3 - (-5) = 8
- Change in y: 6 - (-2) = 8

This indicates that the pentagon has been translated according to the rule (x, y) → (x + 8, y + 8).

### Step 2: Determine Reflection
Next, we need to check if the transformation includes a reflection. We can apply the translation to the original points and then see if they match the transformed coordinates:

1. Applying the translation (x, y) → (x + 8, y + 8) to all the original coordinates gives:
- A(3, 6)
- B(1, 5)
- C(2, 2)
- D(5, 3)
- E(5, 5)

Comparing the new coordinates to the target coordinates:
- A(3, 6): matches A″(3, 6)
- B(-7, -3) → (1, 5): does NOT match B″(5, 5), leading to a difference.
- C(-6, -6) → (2, 2): does NOT match C″(4, 2).
- D(-3, -5) → (5, 3): does NOT match D″(1, 3).
- E(-3, -3) → (5, 5): does NOT match E″(1, 5).

The translation alone does not match.

Let's check with the reflection across the y-axis. A reflection across the y-axis changes the x-coordinates to their opposites.

Reflected A(-5, -2) → A(-(-5), -2) = A(5, -2)
Reflected B(-7, -3) → B(-(-7), -3) = B(7, -3)
Reflected C(-6, -6) → C(-(-6), -6) = C(6, -6)
Reflected D(-3, -5) → D(-(-3), -5) = D(3, -5)
Reflected E(-3, -3) → E(-(-3), -3) = E(3, -3)

### Final Check:
It's evident that after the translation, if we reversed the x-coordinates around the y-axis and applied the correct upward shifts in y, we would relocate them to the desired positions.

### Conclusion:
Upon reviewing options, the correct transformation involving both reflection and translation would be:

**Translated according to the rule (x, y) → (x + 8, y + 2) and reflected across the y-axis.**