Question

Use the image to answer the question.

A coordinate plane shows two triangles of the same shape in quadrants 1 and 2. The horizontal axis ranges from negative 5 to 3 in unit increments. The vertical axis ranges from 0 to 7 in unit increments. Three points labeled upper A upper B upper C form a solid line triangle. The coordinates are upper A at left parenthesis negative 3 comma 5 right parenthesis, upper B at left parenthesis negative 5 comma 1 right parenthesis, and upper C at left parenthesis negative 1 comma 1 right parenthesis. The triangle shows an angle upper B of 63.4 degrees between upper A B and upper B C. A point upper D is marked at left parenthesis 0 comma 2 right parenthesis. Three points labeled upper E upper F upper G form a dotted line triangle. The coordinates are upper E at left parenthesis negative 1 comma 3 right parenthesis, upper F at left parenthesis 3 comma 5 right parenthesis, and upper G at left parenthesis negative 1 comma 7 right parenthesis. The triangle forms an angle upper E of 63.4 degrees between upper E F and upper E G, an angle G of 63.4 degrees between upper E G and upper G F, and an angle upper F of 53.2 degrees between upper G F and upper E F.

Find the measurement of angle A
in the rotation of ΔABC
90°
clockwise around point D
.

Answer 1

To find the measurement of angle \( A \) after a 90° clockwise rotation of triangle \( \Delta ABC \) around point \( D(0, 2) \), we can follow these steps:

1. **Determine the coordinates of points A, B, and C**:
- \( A(-3, 5) \)
- \( B(-5, 1) \)
- \( C(-1, 1) \)

2. **Determine the translation of point \( D \)**:
- The point of rotation \( D \) has coordinates (0, 2).

3. **Perform the 90° clockwise rotation**:
To rotate a point \( (x, y) \) around another point \( (h, k) \) by 90° clockwise, the new coordinates \( (x', y') \) can be calculated as follows:
\[
x' = h + (y - k)
\]
\[
y' = k - (x - h)
\]

4. **Apply the rotation to each point**:

- **Rotate point \( A(-3, 5) \)**:
\[
x' = 0 + (5 - 2) = 3
\]
\[
y' = 2 - (-3 - 0) = 2 + 3 = 5
\]
Thus, \( A' = (3, 5) \).

- **Rotate point \( B(-5, 1) \)**:
\[
x' = 0 + (1 - 2) = -1
\]
\[
y' = 2 - (-5 - 0) = 2 + 5 = 7
\]
Thus, \( B' = (-1, 7) \).

- **Rotate point \( C(-1, 1) \)**:
\[
x' = 0 + (1 - 2) = -1
\]
\[
y' = 2 - (-1 - 0) = 2 + 1 = 3
\]
Thus, \( C' = (-1, 3) \).

5. **Find angle \( A' \) in triangle \( \Delta A'B'C' \)**:
To find the angle \( A' \), we can use the properties of triangles and the fact that a triangle's angles will remain the same after a rotation.

The angle \( A \) in triangle \( \Delta ABC \) is \( 63.4° \). Since the rotation does not change the angle measures, and the shape of the triangle remains the same, angle \( A' \) will also be:
\[
\text{Angle } A' = \text{Angle } A = 63.4°
\]

Therefore, the measurement of angle \( A \) in the rotation of \( \Delta ABC \) \( 90° \) clockwise around point \( D \) is \( \boxed{63.4°} \).