In the adjoining gure, two points A and B lie on the same side of a line XY if AD XY, BEXY and Cis the midpoint of AB, prove that CD CE

To prove that $CD = CE$, we can follow these logical steps:

1. Break Down the Problem

• Understand the given conditions:
  • Points $A$ and $B$ are situated on the same side of line $XY$.
  • Point $C$ is the midpoint of segment $AB$.
  • Line segments $AD$ and $BE$ intersect line $XY$ perpendicularly at points $D$ and $E$, respectively.

2. Relevant Concepts

• Use the properties of midpoints and perpendicular lines.
• Recall that if $C$ is the midpoint of $AB$, then: $AC = CB$
• Use the concept of right triangles formed by the perpendicular segments.

3. Analysis and Detail

• Construct right triangles $ACD$ and $BCE$.
• By the definition of midpoint: $CD = AC \cdot \text{ (the height from C to XY)}$ $CE = BC \cdot \text{ (the height from C to XY)}$
• Since $C$ is the midpoint, $AC = BC$.

4. Verify and Summarize

• Since $D$ and $E$ are on the same line $XY$ and $AD \perp XY$, $BE \perp XY$:
  • Both points $D$ and $E$ are at the same distance from line $XY$, confirming that $CD$ and $CE$ are equal as they are both the heights from $C$ to line $XY$.

Final Answer

Thus, we have proved that $CD = CE$.