Question

If in a triangle ABC, the altitude AM be the bisector of ∠BAD,where D is the mid point of side BC , then prove that (b²−c²)=a²/2.

Answer 1

In triangle ABC, if the altitude AM is the bisector of angle BAD, where D is the midpoint of side BC, we can prove that (b²−c²)=a²/2.

Step-by-step explanation:

In triangle ABC, if the altitude AM is the bisector of angle BAD, where D is the midpoint of side BC, we can prove that (b²−c²)=a²/2.

Here's the step-by-step proof:

1. Using the angle bisector theorem, we know that AD/BD = AC/BC.
2. Since D is the midpoint of BC, we can substitute BD = CD = BC/2.
3. Substituting these values into the previous equation, we get AD = (AC imes BC)/ (BC/2) = 2AC.
4. Using the Pythagorean theorem, we know that a² = b² + c².
5. Substituting the value of AD from step 3 and simplifying, we get b² - c² = (2AC)² - c² = 4(A² - c²).
6. Finally, dividing both sides by 4, we get (b² - c²) = a²/2.