Question

In triangle ABC , let D be the midpoints of BC, if ∠ADB = 45° and ∠ACD = 30° , determine ∠BAD.

Answer 1

Explanation:

Draw BL Perpendicular to AC and join L to D . (See Fig 2.)

Since ∠BCL = 30°, we get ∠CBL = 60°. Since BLC is right-triangle with ∠BCL = 30°, we have BL = BC/2 = BD . Thus in triangle BLD, we observed that BL = BD and ∠DBL = 60°. This implies that BLD is an equilateral triangle and hence LB = LD . Using ∠ LDB = 60° and ∠ADB = 45°, we get ∠ ADL = 15° . But ∠ DAL = 15° . Thus LD = LA. We hence have LD = LA = LB . This implies that L is circumcentre of the triangle BDA . Thus

∠BAD = 1/2∠BLD = 1/2 x 60 = 30°