Question

Use the image to answer the question.

A coordinate plane shows two pentagons crossing all four quadrants. The horizontal axis ranges from negative 2 to 5 in unit increments. The vertical axis ranges from negative 2 to 5 in unit increments. Five points labeled upper A upper B upper C upper D upper E form a solid line pentagon. The unmarked approximate coordinates are as follows: upper A is at left parenthesis 0 comma 5 right parenthesis, upper B is at left parenthesis negative 2 comma 4 right parenthesis, upper C is at left parenthesis negative 1 comma 2 right parenthesis, upper D is at left parenthesis 1.1 comma 3.1 right parenthesis, and upper E is at left parenthesis 1.4 comma 5.2 right parenthesis. Point upper F is marked at left parenthesis 3 comma 3 right parenthesis. Five points labeled upper G upper H upper I upper J upper K form a dotted line pentagon. The unmarked approximate coordinates are as follows: upper G is at left parenthesis 4 comma negative 1 right parenthesis, upper H is at left parenthesis 2.9 comma 1.1 right parenthesis, upper I is at left parenthesis 0.7 comma 1.5 right parenthesis, upper J is at left parenthesis 1 comma 0 right parenthesis, and upper K is at left parenthesis 2 comma negative 2 right parenthesis. No angles are marked.

Which angle corresponds to angle K
in the rotation of pentagon ABCDE
90°
counterclockwise around point F
? Enter the number for the correct angle.

1 = angle A

2 = angle B

3 = angle C

4 = angle D

5 = angle E


(1 point)

Answer 1

To determine which angle corresponds to angle \( K \) in the rotation of pentagon \( ABCDE \) 90° counterclockwise around point \( F \), we can analyze the coordinates and their positions before and after the rotation.

Let's list the points of pentagon \( ABCDE \):
- \( A(0, 5) \)
- \( B(-2, 4) \)
- \( C(-1, 2) \)
- \( D(1.1, 3.1) \)
- \( E(1.4, 5.2) \)

Now, point \( F(3, 3) \) is our center of rotation. When rotating a point \( P(x, y) \) 90° counterclockwise around point \( F(a, b) \), the new coordinates \( P' \) are given by:

\[
P' = (a - (y - b), b + (x - a))
\]

We need to examine the details of the rotation and identify which point of \( ABCDE \) corresponds to point \( K(2, -2) \) of the dotted pentagon.
Let's calculate the new coordinates for each point after rotating 90° counterclockwise around point \( F \).

1. For point \( A(0, 5) \):
\[
A' = (3 - (5 - 3), 3 + (0 - 3)) = (3 - 2, 3 - 3) = (1, 0)
\]

2. For point \( B(-2, 4) \):
\[
B' = (3 - (4 - 3), 3 + (-2 - 3)) = (3 - 1, 3 - 5) = (2, -2)
\]

3. For point \( C(-1, 2) \):
\[
C' = (3 - (2 - 3), 3 + (-1 - 3)) = (3 - (-1), 3 - 4) = (4, -1)
\]

4. For point \( D(1.1, 3.1) \):
\[
D' = (3 - (3.1 - 3), 3 + (1.1 - 3)) = (3 - 0.1, 3 - 1.9) = (2.9, 1.1)
\]

5. For point \( E(1.4, 5.2) \):
\[
E' = (3 - (5.2 - 3), 3 + (1.4 - 3)) = (3 - 2.2, 3 - 1.6) = (0.8, 1.4)
\]

After the rotation, we notice that point \( B' \) has the coordinates \( (2, -2) \), which matches the coordinates of point \( K \).

Thus, angle \( K \) corresponds to angle \( B \).

So the answer is:

**2**