Exercise: Refer to the figure below: Given: AM is an altitude of triangle ABC and AM is a median of triangle ABC. Prove: AB = AC (i.e. you are proving that triangle ABC is isosceles). Line 1: M is the midpoint of BC by definition of midpoint. Line 2: Therefore, HM = KC by definition of midpoint. Line 3: So ∠AMB = ∠AMC by [missing information]. Line 4: Thus, AB = AC because [missing information].
The Correct Answer and Explanation is:
Correct answer:
- Line 3: So ∠AMB = ∠AMC by the definition of altitude (AM is perpendicular to BC, forming equal right angles).
- Line 4: Thus, AB = AC because of the Side-Angle-Side (SAS) congruence theorem, followed by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) principle.
Explanation: To prove that triangle ABC is isosceles, we need to establish that AB is equal to AC. Given that AM is both an altitude and a median of triangle ABC, we will use these properties to demonstrate the required congruence.
First, because AM is a median, it divides segment BC into two equal parts, meaning M is the midpoint of BC. This ensures BM is equal to MC. Additionally, since AM is an altitude, it is perpendicular to BC. This results in right angles at points B and C, giving us ∠AMB = ∠AMC.
Next, we use the Side-Angle-Side (SAS) congruence theorem to prove that triangles AMB and AMC are congruent. The SAS theorem states that if two sides and the included angle of one triangle match two sides and the included angle of another triangle, then the triangles are congruent. In this case:
- AM is shared by both triangles.
- BM equals MC by the definition of a midpoint.
- ∠AMB equals ∠AMC because AM is perpendicular to BC.
Since all three conditions of the SAS theorem are satisfied, triangles AMB and AMC are congruent. Finally, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) principle, AB must equal AC. This confirms triangle ABC is isosceles, completing the proof.
