https://file.aiquickdraw.com/mathbot/file/1726763468573l15mc8l8.jpg

Let's solve the two problems step by step from the image:

### Part (c) - Rotate and Reflect Triangle ΔALT

We are given:
- \( A(-5, -1) \)
- \( L(-3, -2) \)
- \( T(-3, 2) \)

#### Step 1: Rotate 90° clockwise about the origin

To rotate a point 90° clockwise about the origin, we use the transformation rule:
\[
(x, y) \to (y, -x)
\]
So, applying this to each point:
- \( A(-5, -1) \) becomes \( A'( -1, 5 ) \)
- \( L(-3, -2) \) becomes \( L'( -2, 3 ) \)
- \( T(-3, 2) \) becomes \( T'( 2, 3 ) \)

#### Step 2: Reflect over the y-axis

To reflect over the y-axis, we use the rule:
\[
(x, y) \to (-x, y)
\]
So, applying this to each point:
- \( A'( -1, 5 ) \) becomes \( A''( 1, 5 ) \)
- \( L'( -2, 3 ) \) becomes \( L''( 2, 3 ) \)
- \( T'( 2, 3 ) \) becomes \( T''( -2, 3 ) \)

So the coordinates are:
- \( A''( 1, 5 ) \)
- \( L''( 2, 3 ) \)
- \( T''( -2, 3 ) \)

---

### Part (d) - Reflect and Translate Triangle ΔTAB

We are given:
- \( T(2, 3) \)
- \( A(1, 1) \)
- \( B(4, -3) \)

#### Step 1: Reflect over the y-axis

To reflect over the y-axis, we use the same rule:
\[
(x, y) \to (-x, y)
\]
So, applying this to each point:
- \( T(2, 3) \) becomes \( T'( -2, 3 ) \)
- \( A(1, 1) \) becomes \( A'( -1, 1 ) \)
- \( B(4, -3) \) becomes \( B'( -4, -3 ) \)

#### Step 2: Translate by the rule \( (x, y) \to (x+2, y-1) \)

Now, we apply the translation rule to each reflected point:
- \( T'( -2, 3 ) \to T''( 0, 2 ) \)
- \( A'( -1, 1 ) \to A''( 1, 0 ) \)
- \( B'( -4, -3 ) \to B''( -2, -4 ) \)

So the coordinates are:
- \( T''( 0, 2 ) \)
- \( A''( 1, 0 ) \)
- \( B''( -2, -4 ) \)

---

Let me know if you have any questions or want further details on any step.

### Related Questions:
1. How do you reflect a point over the x-axis?
2. What are the general rules for rotating a point 180° and 270°?
3. Can a transformation sequence affect the shape or size of the triangle?
4. How do you perform a reflection over any arbitrary line?
5. What is the difference between translating and rotating a point in a plane?

### Tip:
Always double-check the sign changes during rotations and reflections, as minor mistakes here can lead to incorrect final coordinates.

Rotate and reflect triangle ΔALT, and reflect and translate triangle ΔTAB.

Learn how to perform a 90° clockwise rotation of triangle ΔALT and reflect it over the y-axis. Also, reflect triangle ΔTAB over the y-axis and apply a translation. This guide covers key concepts in rotation, reflection, and translation, suitable for Grades 8-10.

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