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Circle Geometry, SMB-016

The line \(BD\) is a tangent to a circle and the secant \(AD\) intersects the circle at \(A\) and \(C\).
 

Given that  \(AC = 18\)  and  \(CD = 6\), find the value of \(x\).  (2 marks)   

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\(x=12\)

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\(x^2\) \(= 6 \times (18 + 6) \)  
  \(=144\)  
\(x\) \(=12\)  

Circle Geometry, SMB-002

The line \(AT\) is the tangent to the circle at \(A\), and \(BT\) is a secant meeting the circle at \(B\) and \(C\).
  

Given that  \(AT = 12\),  \(BC = 7\)  and  \(CT = x\), find the value of \(x\).  (2 marks)

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\(x = 9\)

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\(\text{Property: square of tangent = product of secant intercepts}\)

\(AT^2\) \(= CT \times BT\)
\(12^2\) \(= x(x + 7)\)
\(144\) \(= x^2 + 7x\)
\(x^2 + 7x-144\) \(= 0\)
\((x + 16)(x-9)\) \(= 0\)

 

\(\therefore x = 9,\  (x \gt 0) \)

Plane Geometry, EXT1 2016 HSC 4 MC

In the diagram, `O` is the centre of the circle `ABC`, `D` is the midpoint of `BC`, `AT` is the tangent at `A` and  `∠ATB = 40^@`.
 

What is the size of the reflex angle `DOA`?

  1. `80^@`
  2. `140^@`
  3. `220^@`
  4. `280^@`
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`C`

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`/_ ODT` `=90^@\ \ text{(line through centre bisecting chord)}`
`/_OAT` `= 90^@\ \ text{(tangent ⊥ to radius at point of contact)}`
`/_ DOA` `= 360-(90 + 90 + 40)`
  `= 140^@`

 

`:. DOA\ \ text{(reflex)}` `= 360-140`
  `= 220^@`

`=>   C`

Plane Geometry, EXT1 2012 HSC 10 MC

The points `A`, `B` and `P` lie on a circle centred at `O`. The tangents to the circle at `A` and `B` meet at the point `T`, and `/_ATB = theta`.

 What is `/_APB` in terms of  `theta`? 

  1. `theta/2`  
  2. `90^@-theta/2`
  3. `theta` 
  4. `180^@-theta` 
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`B`

Show Worked Solution

`/_ BOA= 2 xx /_ APB`

`text{(angles at centre and circumference on arc}\ AB text{)}`

`/_TAO = /_ TBO = 90^@\ text{(angle between radius and tangent)}`

`:.\ theta + /_BOA` `= 180^@\ text{(angle sum of quadrilateral}\ TAOB text{)}`
`theta + 2 xx /_APB` `= 180^@`
`2 xx /_APB` `= 180^@-theta`
`/_APB` `= 90^@-theta/2`

 
`=>  B`