Question

What relections or sequence of reflections transforms figure ABCD into figure EFGH?

E(-5,1)
F(-2,5)
G(-2,8)
H(-4,7)
A(1,-5)
B(5,-2)
C(8,-2)
D(7,-4)

The answer choices are:

Reflect ABCD first over the line x=0, then over the line y=0
Reflect ABCD over the line y=x
Reflect ABCD first over the line x=0 then over the line y=x
Reflect ABCD first over the line y=0 then over the line y=x

Answer 1

Are you sure that's right

Answer 2

To determine which sequence of reflections transforms figure ABCD into figure EFGH, let's first analyze the coordinates of the points in both figures.

**Coordinates of ABCD:**
- A(1, -5)
- B(5, -2)
- C(8, -2)
- D(7, -4)

**Coordinates of EFGH:**
- E(-5, 1)
- F(-2, 5)
- G(-2, 8)
- H(-4, 7)

Now, let's calculate the result of each proposed transformation and see which one results in the points of EFGH.

1. **Reflect over the line \(x = 0\) (the y-axis):**
- A(1, -5) → A'(-1, -5)
- B(5, -2) → B'(-5, -2)
- C(8, -2) → C'(-8, -2)
- D(7, -4) → D'(-7, -4)

Resulting points after first reflection:
- A'(-1, -5)
- B'(-5, -2)
- C'(-8, -2)
- D'(-7, -4)

2. **Next, reflect the resulting points over the line \(y = 0\) (the x-axis):**
- A'(-1, -5) → A''(-1, 5)
- B'(-5, -2) → B''(-5, 2)
- C'(-8, -2) → C''(-8, 2)
- D'(-7, -4) → D''(-7, 4)

Resulting points after second reflection:
- A''(-1, 5)
- B''(-5, 2)
- C''(-8, 2)
- D''(-7, 4)

This result does not match EFGH.

3. **Reflect over the line \(y = x\):**
- A(1, -5) → A'(−5, 1)
- B(5, -2) → B'(−2, 5)
- C(8, -2) → C'(−2, 8)
- D(7, -4) → D'(−4, 7)

Resulting points after reflection over y = x:
- A'(-5, 1)
- B'(-2, 5)
- C'(-2, 8)
- D'(-4, 7)

This result matches EFGH perfectly: E(-5,1), F(-2,5), G(-2,8), H(-4,7).

4. **Reflection over the line \(y = 0\) then over the line \(y = x\):**
- Reflecting A, B, C, D over the y-axis results in A'(-1, -5), B'(-5, -2), C'(-8, -2), D'(-7, -4) (as already calculated).
- Then applying reflection over y=x gives:

- A'(-1, -5) → A''(-5, -1)
- B'(-5, -2) → B''(-2, -5)
- C'(-8, -2) → C''(-2, -8)
- D'(-7, -4) → D''(-4, -7)

This does not match EFGH.

5. **Reflect over y=0 then y=x:**
We already calculated that this does not yield EFGH.

The only successful transformation that matches is **reflecting ABCD over the line \(y = x\)**.

Thus, the answer is:

**Reflect ABCD over the line \(y=x\)**.

Answer 3

Let's carefully re-evaluate the transformation step by step to confirm the accurate reflection that transforms figure \(ABCD\) into figure \(EFGH\).

### Given Points

- **Figure \(ABCD\)**:
- \(A(1, -5)\)
- \(B(5, -2)\)
- \(C(8, -2)\)
- \(D(7, -4)\)

- **Figure \(EFGH\)**:
- \(E(-5, 1)\)
- \(F(-2, 5)\)
- \(G(-2, 8)\)
- \(H(-4, 7)\)

### Analysis

1. **Initial Reflection Calculations**:
Let's explore the transformations one step at a time.

#### Option 1: Reflect over the line \(x = 0\) then \(y = 0\)

- Reflecting over the line \(x=0\) (y-axis):
- \(A(1, -5) \rightarrow A'(-1, -5)\)
- \(B(5, -2) \rightarrow B'(-5, -2)\)
- \(C(8, -2) \rightarrow C'(-8, -2)\)
- \(D(7, -4) \rightarrow D'(-7, -4)\)

- Reflecting those over the line \(y=0\) (x-axis):
- \(A'(-1, -5) \rightarrow A''(-1, 5)\)
- \(B'(-5, -2) \rightarrow B''(-5, 2)\)
- \(C'(-8, -2) \rightarrow C''(-8, 2)\)
- \(D'(-7, -4) \rightarrow D''(-7, 4)\)

Resulting points: \(A''(-1, 5)\), \(B''(-5, 2)\), \(C''(-8, 2)\), \(D''(-7, 4)\) — **This does not match \(EFGH\)**.

#### Option 2: Reflect over the line \(y = x\)

- Reflecting over the line \(y = x\):
- \(A(1, -5) \rightarrow A'(-5, 1)\)
- \(B(5, -2) \rightarrow B'(-2, 5)\)
- \(C(8, -2) \rightarrow C'(-2, 8)\)
- \(D(7, -4) \rightarrow D'(-4, 7)\)

Resulting points: \(A'(-5, 1)\), \(B'(-2, 5)\), \(C'(-2, 8)\), \(D'(-4, 7)\) — **This matches \(EFGH\)**.

#### Other Options

3. **Option 3**: Reflect over \(x=0\) then \(y=x\)

- Reflect over \(x=0\):
- \(A(1, -5) \rightarrow A'(-1, -5)\)
- \(B(5, -2) \rightarrow B'(-5, -2)\)
- \(C(8, -2) \rightarrow C'(-8, -2)\)
- \(D(7, -4) \rightarrow D'(-7, -4)\)

- Reflect those over \(y=x\):
- \(A'(-1, -5) \rightarrow A''(-5, -1)\)
- \(B'(-5, -2) \rightarrow B''(-2, -5)\)
- \(C'(-8, -2) \rightarrow C''(-2, -8)\)
- \(D'(-7, -4) \rightarrow D''(-4, -7)\)

Resulting points: \(A''(-5, -1)\), \(B''(-2, -5)\), \(C''(-2, -8)\), \(D''(-4, -7)\) — **This does not match \(EFGH\)**.

4. **Option 4**: Reflect over \(y=0\) then \(y=x\)

- Over \(y=0\) results in:
- \(A(-1, 5)\), similar to above.

- Then reflect over the line \(y=x\), similar to above, also does not yield \(EFGH\).

### Conclusion

The correct transformation is indeed:
**Reflect \(ABCD\) over the line \(y=x\)**, as this directly transforms the coordinates of \(ABCD\) into those of \(EFGH\). Thank you for your patience!