The altitude, median, and angle bisector from a vertex in an isosceles triangle coincide. True or false?

Prepare for the Geometry CBE Exam with targeted resources. Access interactive quizzes and detailed explanations to master geometric concepts effectively and excel in your test!

Multiple Choice

The altitude, median, and angle bisector from a vertex in an isosceles triangle coincide. True or false?

From the apex of an isosceles triangle (the vertex where the equal sides meet), drop the altitude to the base. The two resulting right triangles are ABH and ACH. Since AB = AC (the equal sides) and AH is common, these two right triangles are congruent (HL congruence). That gives BH = HC, so AH is a median, and it also gives angle BAH = HAC, so AH is an angle bisector. Because AH is defined as the altitude, it is perpendicular to the base as well. Therefore, the altitude, median, and angle bisector from that vertex all lie along the same line AH. This holds for any isosceles triangle, not just equilateral ones or specific vertex angles. Hence the statement is true.

From the apex of an isosceles triangle (the vertex where the equal sides meet), drop the altitude to the base. The two resulting right triangles are ABH and ACH. Since AB = AC (the equal sides) and AH is common, these two right triangles are congruent (HL congruence). That gives BH = HC, so AH is a median, and it also gives angle BAH = HAC, so AH is an angle bisector. Because AH is defined as the altitude, it is perpendicular to the base as well. Therefore, the altitude, median, and angle bisector from that vertex all lie along the same line AH.

This holds for any isosceles triangle, not just equilateral ones or specific vertex angles. Hence the statement is true.

More practice questions