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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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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\(x=12\)
\(x^2\) | \(= 6 \times (18 + 6) \) | |
\(=144\) | ||
\(x\) | \(=12\) |
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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\(x = 9\)
\(\text{Property: square of tangent = product of secant intercepts}\)
\(AT^2\) | \(= CT \times BT\) |
\(12^2\) | \(= x(x + 7)\) |
\(144\) | \(= x^2 + 7x\) |
\(x^2 + 7x-144\) | \(= 0\) |
\((x + 16)(x-9)\) | \(= 0\) |
\(\therefore x = 9,\ (x \gt 0) \)
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`?
`C`
`/_ 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`
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`?
`B`
`/_ 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`