Prove the Following:

Question

If D is the midpoint of BC and AD is perpendicular to BC, prove that triangle ABC is isosceles. Do not use congruent triangle.

Answer 1

Since we aren't supposed to use congruent triangles in the proof, we'll use the Pythagorean Theorem to show that AB = AC. Since two sides of triangle ABC are equal, ABC is isosceles.

1. D is midpoint of BC (Given)
2. AD is perpendicular to BC (Given)
3. BD = CD (Definition of midpoint)
4. ∠CDA is a right angle (Definition of perpendicular)
5. ∠BDA is a right angle (Definition of perpendicular)
6. AC² = CD² + AD² (Pythagorean Theorem)
7. AB² = BD² + AD² (Pythagorean Theorem)
8. AB² = CD² + AD² (Substitution from step 3)
9. AB = AC (Transitive property of equality)
10. ΔABC is isosceles (Definition of Isosceles Triangle)