AB
  with 
A(4,1) and 
B(0,−2) is rotated 
90° counterclockwise about the origin and, then reflected over 
y=x.  Determine the location of the image.  

We need to determine the image of the points \( A(4,1) \) and \( B(0, -2) \) after performing the following transformations:

1. **90° counterclockwise rotation about the origin**
2. **Reflection over the line \( y = x \)**

### Step 1: 90° Counterclockwise Rotation About the Origin
The general rule for rotating a point \( (x, y) \) 90° counterclockwise about the origin is:
\[
(x, y) \rightarrow (-y, x)
\]

#### Apply to Point A(4, 1):
\[
A(4, 1) \rightarrow A'(-1, 4)
\]

#### Apply to Point B(0, -2):
\[
B(0, -2) \rightarrow B'(2, 0)
\]

### Step 2: Reflection Over the Line \( y = x \)
The rule for reflecting a point over the line \( y = x \) is:
\[
(x, y) \rightarrow (y, x)
\]

#### Apply to Point A'(-1, 4):
\[
A'(-1, 4) \rightarrow A''(4, -1)
\]

#### Apply to Point B'(2, 0):
\[
B'(2, 0) \rightarrow B''(0, 2)
\]

### Final Answer
- The image of point \( A(4, 1) \) is \( A''(4, -1) \).
- The image of point \( B(0, -2) \) is \( B''(0, 2) \).

Thus, the new locations of the points after the transformations are \( A''(4, -1) \) and \( B''(0, 2) \).

Would you like further clarification or more details? Here are 5 questions to extend your understanding:

1. What is the formula for rotating a point by 180° about the origin?
2. How does the reflection change when reflecting over the x-axis instead of the line \( y = x \)?
3. How do you find the midpoint of a line segment between two points?
4. What is the effect of a 90° clockwise rotation on a point?
5. How can you verify the accuracy of a transformation using matrices?

**Tip:** When performing multiple transformations, always follow the order of operations step by step to avoid confusion.

AB with A(4,1) and B(0,−2) is rotated 90° counterclockwise about the origin and, then reflected over y=x. Determine the location of the image.

Learn how to find the new location of points A(4,1) and B(0,−2) after a 90° counterclockwise rotation about the origin and a reflection over the line y=x. Understand key concepts in coordinate geometry and transformations, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation