Question

Which red triangle shows a 90° counterclockwise rotation of the blue triangle? Check all that apply.

On a coordinate plane, a blue triangle has points (negative 1, 4), (negative 5, 4), (negative 1, 1). A red triangle has points (negative 4, negative 1), (negative 1, negative 1), (negative 4, negative 5).
On a coordinate plane, a blue triangle has points (1, 1), (4, 5), (4, 1). A red triangle has points (negative 1, 1), (negative 1, 4), (negative 5, 4).
On a coordinate plane, a blue triangle has points (0, 1), (4, 4), (4, 1). A red triangle has points (negative 4, 1), (negative 4, 4), (0, 1).
On a coordinate plane, a blue triangle has points (negative 5, 4), (negative 1, 4), (negative 1), 1). A red triangle has points (1, 1), (4, 1), (4, 5).
On a coordinate plane, a blue triangle has points (negative 2, 5), (4, 5), (4, 1). A red triangle has points (negative 1, 4), (negative 5, 4), (negative 5, negative 2).

Answer 1

To find which red triangle shows a 90° counterclockwise rotation of the blue triangle, we can use the rotation formula for points in a coordinate plane. The rule for a 90° counterclockwise rotation about the origin is:

- For a point \((x, y)\), the new point after rotation will be \((-y, x)\).

Let's apply this formula to the coordinates of each blue triangle provided.

1. **Blue Triangle Points:**
\[
A(-1, 4), B(-5, 4), C(-1, 1)
\]

**Rotated Points:**
\[
A'(-4, -1), B'(-4, -5), C'(-1, -1)
\]

**Red Triangle Points:**
\[
A'(-4, -1), B'(-1, -1), C'(-4, -5) \quad \text{(This matches)} \Rightarrow \text{This is a match!}
\]

2. **Blue Triangle Points:**
\[
A(1, 1), B(4, 5), C(4, 1)
\]

**Rotated Points:**
\[
A'(-1, 1), B'(-5, 4), C'(-1, 4)
\]

**Red Triangle Points:**
\[
A'(-1, 1), B'(-1, 4), C'(-5, 4) \quad \text{(This does not match)}
\]

3. **Blue Triangle Points:**
\[
A(0, 1), B(4, 4), C(4, 1)
\]

**Rotated Points:**
\[
A'(-1, 0), B'(-4, 4), C'(-4, 1)
\]

**Red Triangle Points:**
\[
A'(-4, 1), B'(-4, 4), C'(0, 1) \quad \text{(This does not match)}
\]

4. **Blue Triangle Points:**
\[
A(-5, 4), B(-1, 4), C(-1, 1)
\]

**Rotated Points:**
\[
A'(-4, -5), B'(-4, -1), C'(1, -1)
\]

**Red Triangle Points:**
\[
A'(1, 1), B(4, 1), C(4, 5) \quad \text{(This does not match)}
\]

5. **Blue Triangle Points:**
\[
A(-2, 5), B(4, 5), C(4, 1)
\]

**Rotated Points:**
\[
A'(-5, -2), B'(-5, 4), C'(-1, 4)
\]

**Red Triangle Points:**
\[
A'(-1, 4), B'(-5, 4), C'(-5, -2) \quad \text{(This does not match)}
\]

**Conclusion:**
The only red triangle that correctly shows a 90° counterclockwise rotation of the blue triangle is the first one.

**Answer: The first red triangle matches.**