Question

AOB is a diameter of the circle, C, D, E are any three points on the semi circle. find angle ACD+BED

Answer 1

AOB is a diameter of the circle, C, E, D are three points on the circle.

To find: ∠ACD +∠BED
Construction: Join CO, DO and EO

Assume AC = CD = AO
Assume DE = EB

So In ΔACO
It is an equilateral triangle, Hence ∠ACO = 60°

In ΔCDO
It is an equilateral triangle, Hence ∠DCO = 60°


∠DOE =∠EOB (as equal chord subtend equal angle at the centre as DE = EB
As ∠AOB = 180°
So 180° = ∠AOC +∠COD +∠DOE +∠EOB
And ∠AOC =∠COD = 60°
And ∠DOE = ∠EOB
Hence 2∠DOE = 180° - 60° -60°
∠DOE = 30°

In ΔODE
∠ODE =∠OED (isosceles triangle)
And ∠ODE + ∠OED + ∠DOE = 180°
So 2∠ODE = 180°-30° = 150°
So ∠ODE = 75°

In ΔOEB
∠OEB =∠OBE (isosceles triangle)
And ∠OEB + ∠OEB + ∠BOE = 180°
So 2∠OEB = 180°-30° = 150°
So ∠OEB = 75°

So ∠ACD +∠BED = ∠ACO +∠DCO + ∠DEO +∠BEO = 60° + 60° + 75° + 75° = 270°