Give me some problems on a graph that has me dialate, transform,translate ,rotate shapes across the graph  with answers 

Here are some problems involving dilation, transformation, translation, and rotation of shapes on a coordinate plane. Each problem is followed by its solution.

### Problem 1: Dilation
**Problem:**  
Dilate triangle \( A(1,1), B(3,1), C(2,3) \) by a scale factor of 2, centered at the origin.

**Solution:**  
To dilate a point \( (x, y) \) by a scale factor of \( k \) centered at the origin, multiply both coordinates by \( k \).

- New \( A' = (1 \times 2, 1 \times 2) = (2, 2) \)
- New \( B' = (3 \times 2, 1 \times 2) = (6, 2) \)
- New \( C' = (2 \times 2, 3 \times 2) = (4, 6) \)

So, the new triangle is \( A'(2, 2), B'(6, 2), C'(4, 6) \).

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### Problem 2: Translation
**Problem:**  
Translate quadrilateral \( P(2,1), Q(4,1), R(4,3), S(2,3) \) by \( (x+3, y-2) \).

**Solution:**  
To translate a point, add 3 to the x-coordinate and subtract 2 from the y-coordinate.

- New \( P' = (2 + 3, 1 - 2) = (5, -1) \)
- New \( Q' = (4 + 3, 1 - 2) = (7, -1) \)
- New \( R' = (4 + 3, 3 - 2) = (7, 1) \)
- New \( S' = (2 + 3, 3 - 2) = (5, 1) \)

So, the new quadrilateral is \( P'(5, -1), Q'(7, -1), R'(7, 1), S'(5, 1) \).

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### Problem 3: Reflection
**Problem:**  
Reflect triangle \( A(1,2), B(3,4), C(5,2) \) across the y-axis.

**Solution:**  
To reflect across the y-axis, change the sign of the x-coordinate.

- New \( A' = (-1, 2) \)
- New \( B' = (-3, 4) \)
- New \( C' = (-5, 2) \)

So, the reflected triangle is \( A'(-1, 2), B'(-3, 4), C'(-5, 2) \).

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### Problem 4: Rotation
**Problem:**  
Rotate rectangle \( P(1, 1), Q(4, 1), R(4, 3), S(1, 3) \) by 90° counterclockwise around the origin.

**Solution:**  
To rotate a point 90° counterclockwise, use the transformation \( (x, y) \to (-y, x) \).

- New \( P' = (-1, 1) \)
- New \( Q' = (-1, 4) \)
- New \( R' = (-3, 4) \)
- New \( S' = (-3, 1) \)

So, the rotated rectangle is \( P'(-1, 1), Q'(-1, 4), R'(-3, 4), S'(-3, 1) \).

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### Problem 5: Dilation and Translation
**Problem:**  
Dilate triangle \( T(2,2), U(4,2), V(3,5) \) by a scale factor of \( \frac{1}{2} \) centered at the origin, and then translate it by \( (x+1, y+3) \).

**Solution:**  
Step 1: Dilation by \( \frac{1}{2} \) (multiply coordinates by \( \frac{1}{2} \)):

- New \( T' = (2 \times \frac{1}{2}, 2 \times \frac{1}{2}) = (1, 1) \)
- New \( U' = (4 \times \frac{1}{2}, 2 \times \frac{1}{2}) = (2, 1) \)
- New \( V' = (3 \times \frac{1}{2}, 5 \times \frac{1}{2}) = (1.5, 2.5) \)

Step 2: Translation by \( (x+1, y+3) \):

- New \( T'' = (1 + 1, 1 + 3) = (2, 4) \)
- New \( U'' = (2 + 1, 1 + 3) = (3, 4) \)
- New \( V'' = (1.5 + 1, 2.5 + 3) = (2.5, 5.5) \)

So, the final triangle is \( T''(2, 4), U''(3, 4), V''(2.5, 5.5) \).

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Would you like further details on any of these transformations or another type of transformation problem?

### 5 Related Questions:
1. How do you determine the center of rotation when not given explicitly?
2. What happens when you reflect a shape across both the x-axis and y-axis?
3. How do you perform a rotation of a shape by 180°?
4. How does changing the center of dilation affect the resulting image?
5. Can you translate a shape along a curved path? If so, how would that affect its coordinates?

### Tip:
When working with dilations and transformations, it is helpful to visualize the graph to better understand how the points move across the plane.

Give me some problems on a graph that has me dilate, transform, translate, rotate shapes across the graph with answers.

Practice graph transformations, including dilation, translation, rotation, and reflection of shapes. Learn how to apply key concepts in coordinate geometry with step-by-step solutions. Suitable for Grades 8-10 students.

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