The problem involves finding a glide reflection that maps the triangle $\triangle ABC$ with vertices $A = (4, 2)$, $B = (6, 2)$, and $C = (4, 5)$ to a new triangle $\triangle A'B'C'$ with vertices $A' = (1, -5)$, $B' = (3, -5)$, and $C' = (1, -8)$. The glide reflection consists of a reflection across a line followed by a translation along the vector $\vec{j} = [a, 1]$.

Step 1: Translation

First, let’s find the vector between corresponding points $A \to A'$, $B \to B'$, and $C \to C'$.

• For $A(4, 2) \to A'(1, -5)$: the vector is $(1 - 4, -5 - 2) = (-3, -7)$.
• For $B(6, 2) \to B'(3, -5)$: the vector is $(3 - 6, -5 - 2) = (-3, -7)$.
• For $C(4, 5) \to C'(1, -8)$: the vector is $(1 - 4, -8 - 5) = (-3, -13)$.

These vectors indicate that the points are not simply translated by the same vector, but there’s a combination of a reflection and a translation.

Step 2: Identify Translation Vector

The translation vector $\vec{j} = [a, 1]$. From the given, we know that the vertical component of the translation is 1. Now we need to calculate the horizontal component $a$.

If we subtract the vertical component from the total vertical displacement between the points $A \to A'$, which is -7, then the remainder must come from the reflection. Hence, the translation accounts for 1 unit of the total -7 displacement, leaving -6 units for the reflection. Similarly, the horizontal displacement is -3, so part of this comes from the translation and part from the reflection.

Step 3: Find the Line of Reflection

To find the line of reflection, we know that it must be equidistant between corresponding points of the triangles. Specifically, it must lie halfway between points along a perpendicular to the translation vector.

The midpoints between corresponding vertices are:

• Midpoint between $A(4, 2)$ and $A'(1, -5)$: $\left( \frac{4 + 1}{2}, \frac{2 - 5}{2} \right) = \left( \frac{5}{2}, \frac{-3}{2} \right) = (2.5, -1.5)$.
• Midpoint between $B(6, 2)$ and $B'(3, -5)$: $\left( \frac{6 + 3}{2}, \frac{2 - 5}{2} \right) = \left( \frac{9}{2}, \frac{-3}{2} \right) = (4.5, -1.5)$.
• Midpoint between $C(4, 5)$ and $C'(1, -8)$: $\left( \frac{4 + 1}{2}, \frac{5 - 8}{2} \right) = \left( \frac{5}{2}, \frac{-3}{2} \right) = (2.5, -1.5)$.

The midpoints show that the line of reflection must be horizontal and pass through $y = -1.5$.

Thus, the line of reflection is: $y = -1.5$

Step 4: Solve for $a$

Since the translation vector is $[a, 1]$, and the horizontal displacement is -3, the reflection must account for part of this displacement. The translation contributes $a$, so the reflection must contribute $-3 - a$. Since the reflection reflects points across the line, we have: $-3 = 2a$ Solving for $a$, we get: $a = -1.5$

Final Answer

• The translation vector is $\vec{j} = [-1.5, 1]$.
• The line of reflection is $y = -1.5$.

Would you like any further details or clarifications?

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Here are 5 related questions:

1. How do you determine the line of reflection in a general transformation?
2. Can any translation vector be used in a glide reflection, or are there restrictions?
3. How can glide reflections be used to solve real-world geometric problems?
4. What is the difference between a glide reflection and a simple reflection?
5. How can you find the equation of a reflection line given different sets of points?

Tip: In glide reflections, the translation vector often helps determine key parts of the reflection line and vice versa. Understanding the relationship between translation and reflection is essential for mastering these transformations.