https://file.aiquickdraw.com/mathbot/file/17272661867615rcnlwuk.jpg

Let's break down the steps to solve this transformation problem:

1. **Translation** by \( T\left(-1, 2\right) \):
   Each point \( (x, y) \) will be translated by moving \( x \) to \( x - 1 \) and \( y \) to \( y + 2 \).

   - \( P(2, -6) \) → \( P_1(2 - 1, -6 + 2) = P_1(1, -4) \)
   - \( Q(-3, 5) \) → \( Q_1(-3 - 1, 5 + 2) = Q_1(-4, 7) \)
   - \( R(-1, -4) \) → \( R_1(-1 - 1, -4 + 2) = R_1(-2, -2) \)

2. **Reflection** across the line \( x = 2 \):
   To reflect a point \( (x, y) \) across the line \( x = 2 \), we apply the transformation:
   \[
   x' = 2 + (2 - x) = 4 - x
   \]
   
   - \( P_1(1, -4) \) → \( P_2(4 - 1, -4) = P_2(3, -4) \)
   - \( Q_1(-4, 7) \) → \( Q_2(4 - (-4), 7) = Q_2(8, 7) \)
   - \( R_1(-2, -2) \) → \( R_2(4 - (-2), -2) = R_2(6, -2) \)

3. **Rotation** by \( 90^\circ \) counterclockwise around the origin \( O(0, 0) \):
   The rotation formula for a \( 90^\circ \) counterclockwise rotation is:
   \[
   x' = -y \quad \text{and} \quad y' = x
   \]
   
   - \( P_2(3, -4) \) → \( P'(4, 3) \)
   - \( Q_2(8, 7) \) → \( Q'(-7, 8) \)
   - \( R_2(6, -2) \) → \( R'(2, 6) \)

Now, comparing the transformed coordinates to the answer choices:
- The correct answer is **D**: \( P'(8, 3), Q'(-3, -8), R'(6, 6) \).

Do you need any further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you derive reflection formulas for lines other than \( x = 2 \)?
2. What happens if you reflect a point over the \( y \)-axis instead?
3. How do rotations change when the angle is different from \( 90^\circ \)?
4. What are the general transformation rules for scaling?
5. How can you determine the center of rotation for arbitrary points?

**Tip:** Always apply transformations step-by-step to avoid confusion and ensure accuracy!

Titik P(2, -6), Q(-3, 5), dan R(-1, -4) ditranslasi oleh T(-1, 2) dilanjutkan refleksi oleh garis x = 2, kemudian dirotasi sejauh 90° berlawanan arah jarum jam dengan pusat O. Bayangan ketiga titik tersebut adalah...

Learn how to solve transformation problems involving translation by (-1, 2), reflection across x = 2, and a 90° counterclockwise rotation around the origin. This problem involves key concepts in coordinate geometry and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation