Question

3. In ∆ABC, AD ⊥ BC and AD bisects ∠A. Prove that ∆ABD ≅ ∆ACD.

Also, prove that ∆ABC is isoscele
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Answer 1

Answer

Given:

In ΔABC, angle C is an obtuse angle and AD is perpendicular to BC.

Furthermore, we also have that AB2 = AC2 + 3BC2.

We need to show that BC = CD.

Based on the information provided, a diagram of the said triangle is constructed below.

In ΔABC, angle C is obtuse.

Thus, the Obtuse Angle Theorem states that

AB2 = AC2 + BC2 + 2BC × CD ....... (1)

Combining Equation (1) with the given equation AB2 = AC2 + 3BC2, we get

AC2 + 3BC2 = AC2 + BC2 + 2BC × CD

2BC² = 2 BC × CD

BC = DC

Therefore, given the provided information about ΔABC, we can see that BC = DC.