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 $A'(2, 2), B'(6, 2), C'(4, 6)$.

---

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 $P'(5, -1), Q'(7, -1), R'(7, 1), S'(5, 1)$.

---

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 $A'(-1, 2), B'(-3, 4), C'(-5, 2)$.

---

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 $P'(-1, 1), Q'(-1, 4), R'(-3, 4), S'(-3, 1)$.

---

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 $T''(2, 4), U''(3, 4), V''(2.5, 5.5)$.

---

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.