https://file.aiquickdraw.com/mathbot/file/1728400174826h4iwvw9h.jpg

The image shows several triangles on a coordinate grid along with two questions related to transformations:

1. **(a) Triangle \( \text{ABC} \) is mapped into triangle \( A_1B_1C_1 \) by a reflection. Find the equation of the axis of reflection.**

2. **(b) Triangle \( \text{ABC} \) is mapped onto triangle \( A_2B_2C_2 \) by a certain single transformation. Describe fully the single transformation.**

### Analysis and Solution

**(a) Finding the Equation of the Axis of Reflection**

- The original triangle \( \text{ABC} \) has vertices \( A(0, 2) \), \( B(-2, 0) \), and \( C(-1, -1) \).
- The reflected triangle \( A_1B_1C_1 \) has vertices \( A_1(0, -2) \), \( B_1(-2, 0) \), and \( C_1(-1, 1) \).

The axis of reflection should be the line that is equidistant to corresponding points of \( \text{ABC} \) and \( A_1B_1C_1 \). This axis is the horizontal line through \( y = 0 \).

**Equation of the axis of reflection:**
\[
y = 0
\]

**(b) Describing the Single Transformation**

- Triangle \( \text{ABC} \) is mapped onto triangle \( A_2B_2C_2 \) by a translation.
- The translation can be found by comparing corresponding points, e.g., \( A(0, 2) \) maps to \( A_2(0, -4) \).
- This implies a vertical shift of 6 units downwards.

**Description of the transformation:**
- **Translation** 6 units downwards.

---

Would you like further details or have any questions? Here are some related questions you might want to explore:

1. How can I find the equation of the axis of symmetry for other transformations?
2. How is a reflection different from a rotation in terms of properties?
3. Can a transformation involve a combination of reflection and translation?
4. How would you write the matrix representation for a reflection over \( y = 0 \)?
5. What is the difference between a translation and a glide reflection?

**Tip:** When identifying transformations, always look at how the coordinates change between the original and transformed shapes.

The graph shows triangles ABC, A₁B₁C₁, A₂B₂C₂, A₃B₃C₃, and A₄B₄C₄. Use the graph to answer the following questions: (a) Triangle ABC is mapped into triangle A₁B₁C₁ by a reflection. Find the equation of the axis of reflection. (b) Triangle ABC is mapped onto triangle A₂B₂C₂ by a certain single transformation. Describe fully the single transformation.

Analyze geometric transformations of triangles on a coordinate grid. This problem explores reflection with line equations and vertical translation using specific points, making it ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation