Question

Please assist quickly! 100 points, no rude answers without report!

Answer 1

The correct answer is option c.

You can see the explanations in the attachment.

Answer 2

B) A reflection over L maps point C to point A and maps point D to itself. Reflections preserve distances, so D must be equidistant from A and C.

Explanation:

We are given the following information:

• Line
  `\ell` is the perpendicular bisector of segment
  `\overline{\sf AC}`. This means that line
  `\ell` passes through the midpoint of segment
  `\overline{\sf AC}`, and it is perpendicular to
  `\overline{\sf AC}`, splitting
  `\overline{\sf AC}` into two equal parts.
• D is any point on line
  `\ell`.

We want to prove that point D is equidistant from points A and C, which means that the distance from D to A is the same as the distance from D to C. To do this, we can consider a reflection over line
`\ell`.

When we perform a reflection over a line, it transforms objects on one side of the line to corresponding objects on the other side of the line while preserving distances along the line of reflection.

So, when we perform a reflection over line
`\ell`:

• Point C is mapped to point A.
• Point D remains where it is.

Now, since reflections preserve distances along the line of reflection, the distance from D to A must be the same as the distance from D to C. In other words, D is equidistant from A and C.

Therefore, the correct answer is option B:

• A reflection over
  `\ell` maps point C to point A and maps point D to itself. Reflections preserve distances, so D must be equidistant from A and C.

To summarize, line
`\ell` is the perpendicular bisector of segment
`\overline{\sf AC}`. When we reflect over line
`\ell`, it swaps the positions of A and C while keeping point D in its place. Since reflections preserve distances, D must be equidistant from the new positions of A and C, which are the same as the original positions of A and C. Therefore, D is equidistant from A and C.