The problem involves a triangle reflected across a vertical line, and we're asked to find:

1. The coordinates of the other vertex of the reflected triangle.
2. The equation of the line of reflection.

Analyzing the Reflection

Given:

• The original triangle has vertices (2, 3), (2, 7), and (4, 5).
• The reflected triangle has two vertices at (7, 3) and (7, 7).

Step 1: Identifying the Line of Reflection

Since the reflection is over a vertical line, it must be equidistant between the original and reflected x-coordinates of the given points.

1. For the point (2, 3) reflecting to (7, 3), the line of reflection lies halfway between x = 2 and x = 7. $\text{Midpoint x-coordinate} = \frac{2 + 7}{2} = 4.5$
2. Thus, the line of reflection is $x = 4.5$.

Step 2: Finding the Missing Vertex of the Reflected Triangle

The third vertex of the original triangle is at (4, 5). Reflecting this point across $x = 4.5$:

• The distance from x = 4 to x = 4.5 is 0.5 units.
• Reflecting it across this line would move it 0.5 units to the right of x = 4.5, placing it at $x = 5$.

Thus, the reflected third vertex is at $(5, 5)$.

Answers

(a) The coordinates of the other vertex of the reflected triangle are: $(5, 5)$

(b) The equation of the line of reflection is: $x = 4.5$

Would you like more details or have any questions?

Related Questions:

1. What happens if the line of reflection is horizontal instead of vertical?
2. How would the coordinates change if the reflection line was at $x = 3$?
3. Can you explain how to reflect points over lines other than vertical or horizontal?
4. What if we wanted to rotate the triangle instead of reflecting it?
5. How do reflections affect the area of a triangle?

Tip:

When reflecting a point over a vertical or horizontal line, calculate the midpoint between the point and its image to determine the line of reflection.