Circle Geometry, SMB-017

In the diagram, the points `A`, `B`, `C` and `D` are on the circumference of a circle, whose centre `O` lies on `BD`. The chord `AC` intersects the diameter `BD` at `Y`. The tangent at `D` passes through the point `X`.

It is given that `∠CYB = 100^@` and `∠DCY = 30^@`.

What is the size of `∠ACB`? (1 mark)

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What is the size of `∠CBD`? (2 marks)

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Show Answers Only

`60^@`
`20^@`

Show Worked Solution

i. `∠DCB`	`= 90^@\ \ text{(angle in semi-circle)}`
`∠ACB`	`= 90-30`
	`= 60^@`

ii. `∠CYD = 180-100=80^@\ \ text{(180° in straight line)}`

`∠CDY = 180-(80+30)=70^@\ \ text{(180° in Δ)}`

`∠CBD = 180-(90+70)=20^@\ \ text{(180° in Δ)}`