Question

Complete the equations identifying both lines of reflection that will flip the given figure onto itself.
Coordinates are A(3,-4) B(3,5) C(5,-2) D(-4,-2)
(1 point) The lines of reflection are x= and y= .

Answers

Answered by GPT-4o mini
To identify the lines of reflection that will flip the figure formed by the points A(3,-4), B(3,5), C(5,-2), and D(-4,-2) onto itself, we can analyze the coordinates of the points.

1. **Finding the lines of reflection:**

- **Vertical Line of Reflection:**

The x-coordinates of points A and B are both 3, which indicates symmetry about a vertical line. The coordinates of points C and D suggest they reflect over a vertical line at x = 0, as point D's x-coordinate is -4 and point C's x-coordinate is 5.

To find the x-coordinate of the vertical line of reflection, we can average the x-values of points that are pairs. Upon observing the x-coordinates:
- Reflecting A(3, -4) and B(3, 5) would have their mid-point between them at x = 3.
- Reflecting C(5, -2) and D(-4, -2) would have their mid-point found by averaging both x-coordinates: (5 + (-4))/2 = 0.5.

Observing symmetry, the best line of reflection here is a vertical line passing through the x = 0. Hence:

\( x = 0 \).

- **Horizontal Line of Reflection:**

To find a horizontal line, we look at the y-coordinates. The y-coordinates of points A and D are at -4 and -2 respectively; similarly for points B and C. The points will have a mid-line reflecting around their averages.

Looking closely:
- A(3, -4) and C(5, -2) can be averaged; (-4 + (-2))/2 = -3.
- C(5, -2) and D(-4, -2) average at -2 (as both share the same y-coordinate which contributes to the same horizontal reflection).

Thus, the average between point A and point B provides a y flip hence the interesting part to conclude here is noting this y-value positions for reflection.

The horizontal reflection can then be confirmed on y = 1/2.

2. **Finalizing the reflection lines:**

Putting these together, the lines of reflection for this figure are:
- \( x = 0 \) (vertical line of reflection)
- \( y = 1/2 \) (horizontal line of reflection)

So your final answer would be:
The lines of reflection are \( x = 0 \) and \( y = 1/2 \).
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