https://file.aiquickdraw.com/mathbot/file/1727355707024bx6ewxfn.jpg

Let me go through the questions in the image and solve them step by step.

### Question 7:
Translation of points \( A(4, 4) \) and \( B(2, -4) \) by the vector \( T = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \).

The formula for translation is:
\[
A'(x', y') = (x + 3, y + 5)
\]
Let's apply this:

- For \( A(4, 4) \):
\[
A'(x', y') = (4 + 3, 4 + 5) = (7, 9)
\]
- For \( B(2, -4) \):
\[
B'(x', y') = (2 + 3, -4 + 5) = (5, 1)
\]

So the new coordinates are:
\[
A'(7, 9) \quad \text{and} \quad B'(5, 1)
\]

Thus, the correct option for **Question 7** is **a.**.

---

### Question 8:
Reflection of point \( P(5, 6) \) with respect to the line \( y = x \).

For a reflection over the line \( y = x \), the formula is:
\[
P'(x', y') = (y, x)
\]
Simply swap the coordinates:

For \( P(5, 6) \), the reflection is:
\[
P'(x', y') = (6, 5)
\]

Thus, the correct option for **Question 8** is **d.**.

---

### Question 9:
Rotation of triangle \( ABC \) with vertices \( A(2, 2) \), \( B(4, 5) \), and \( C(6, -1) \) by \( 90^\circ \) counterclockwise around the origin \( O(0, 0) \).

The formula for a \( 90^\circ \) counterclockwise rotation is:
\[
A'(x', y') = (-y, x)
\]
Apply this for each vertex:

- For \( A(2, 2) \):
\[
A'(x', y') = (-2, 2)
\]
- For \( B(4, 5) \):
\[
B'(x', y') = (-5, 4)
\]
- For \( C(6, -1) \):
\[
C'(x', y') = (1, 6)
\]

Thus, the new coordinates are:
\[
A'(-2, 2), \quad B'(-5, 4), \quad C'(1, 6)
\]

The correct option for **Question 9** is **c.**.

---

### Summary of Correct Answers:
- Question 7: **a.**
- Question 8: **d.**
- Question 9: **c.**

Let me know if you'd like further details or clarifications!

---

### Related questions:
1. How do you reflect points over lines other than \( y = x \)?
2. What is the general formula for a translation by a vector \( T(x, y) \)?
3. How do you rotate points around a different center of rotation, not \( (0, 0) \)?
4. What is the effect of a \( 180^\circ \) rotation on points?
5. Can you explain the reflection of a triangle over the \( x \)-axis?

### Tip:
When rotating a point by \( 90^\circ \) counterclockwise, swap the coordinates and negate the new x-coordinate!

Translation of points A(4, 4) and B(2, -4) by the vector T = [3, 5]; Reflection of point P(5, 6) over the line y = x; Rotation of triangle ABC with vertices A(2, 2), B(4, 5), and C(6, -1) by 90° counterclockwise around the origin.

Learn how to translate points, reflect over y=x, and rotate a triangle by 90° counterclockwise. This example covers translation of points A(4, 4) and B(2, -4) by vector [3, 5], reflection of P(5, 6) over the line y=x, and rotation of triangle ABC.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation