Question

Use the given information to prove the following theorem.

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
We let be any point on line , but different from point .

Answer 1

Let's proof

PQ is the perpendicular bisector Hence

• CQ=DQ(Bisected sides)

Now apply Pythagorean theorem

`\\ \tt\hookrightarrow PQ^2+QD^2=PD^2`--(1)

`\\ \tt\hookrightarrow PQ^2+CQ^2=PC^2`

As QD=CD

`\\ \tt\hookrightarrow PQ^2+QD^2=PC^2`--(2)

From (1) and (2)

`\\ \tt\hookrightarrow PC^2=PD^2`

`\\ \tt\hookrightarrow PC=PD`

Answer 2

Given
`\overline{\rm PQ}` is the
`\perp` bisector of
`\overline{\rm CD}`

⇒
`\overline{\rm CQ}=\overline{\rm CD}`

⇒ ΔPQD ≅ ΔPQC

⇒ CP = PD

Explanation:

Given
`\overline{\rm PQ}` is the
`\perp` bisector of
`\overline{\rm CD}`

⇒
`\overline{\rm CQ}=\overline{\rm CD}`

⇒ ΔPQD ≅ ΔPQC

⇒ CP = PD