Question

c is the centroid of isosceles triangle abd with vertex angle ∠abd. does the following proof correctly justify that triangles abe and dbe are congruent?

1. it is given that triangle abd is isosceles, so segment ab is congruent to db by the definition of isosceles triangle.
2. triangles abe and dbe share side be, so it is congruent to itself by the reflexive property.
3. it is given that c is the centroid of triangle abd, so segment be is a perpendicular bisector.
4. e is a midpoint, creating congruent segments ae and de, by the definition of midpoint.
5. triangles abe and dbe are congruent by the sss postulate.

image of triangle abd with centroid c and midpoint e on ad, be connecting b to e
options:

• there is an error in line 1; segments ab and bc are congruent.
• there is an error in line 2; segment be is not a shared side.
• there is an error in line 3; segment be should be a median
• the proof is correct.

Explanation:

1. Line 1 is correct: By definition, isosceles triangle ABD with vertex ∠ABD has $\overline{AB} \cong \overline{DB}$.
2. Line 2 is correct: $\overline{BE}$ is a common side to both triangles, so it is congruent to itself via the reflexive property.
3. Line 3 is incorrect: A centroid is the intersection of a triangle's medians. While in an isosceles triangle, the median from the vertex angle is also a perpendicular bisector, the direct property of the centroid only confirms BE is a median (not a perpendicular bisector) initially. The proof incorrectly overstates the property here.
4. While E being a midpoint (from BE being a median) is valid, the error in line 3 makes the proof's reasoning flawed at that step.

Answer:

There is an error in line 3; segment BE should be a median