Question

Given: CD is the perpendicular bisector of AB.

Prove: Point C is equidistant from points A and B.
A) It is given that CD is the perpendicular bisector of AB.
B) Since CD is perpendicular to AB, AE ≅ EB.
C) By definition of perpendicular lines, ∠AEC and ∠BEC are right angles.
D) Since all right angles are congruent, ∠AEC ≅ ∠BEC.
E) Draw AC and BC.
F) ∆AEC ≅ ∆BEC by the side-angle-side congruence postulate.
G) Corresponding parts of congruent triangles are congruent, so AC ≅ BC.
H) It can be concluded that point C is equidistant from points A and B.

In which step was the error made and how should the error be corrected?

Answer 1

The error is made in Step E, which assumes that drawing AC and BC proves that AC and BC are congruent. A different approach is needed to prove that point C is equidistant from points A and B.

The error is made in Step E. Drawing AC and BC does not necessarily prove that AC and BC are congruent.

This step assumes that triangle AEC and triangle BEC are congruent based on the fact that CD is the perpendicular bisector of AB, but this is not a valid assumption.

To correct the error, we need to use a different method to prove that point C is equidistant from points A and B. One possible approach is to show that angle ACE is congruent to angle BCE, using the fact that CD is the perpendicular bisector of AB.

Then, we can use angle-side-angle congruence to prove triangle ACE congruent to triangle BCE.

Finally, we can conclude that AC is congruent to BC, which means point C is equidistant from points A and B.