Higher June 2018 Paper 1 Q11
11
\(A\) and \(B\) are points on a circle, centre \(O\).
\(BC\) is a tangent to the circle.
\(AOC\) is a straight line.
Angle \(ABO = x^\circ\).
Find the size of angle \(ACB\), in terms of \(x\).
Give your answer in its simplest form.
Give reasons for each stage of your working. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(90 - 2x\) | M1 | for identifying an unknown angle eg \(BAO = x\), \(AOB = 180 - 2x\), \(OBC = 90\), \(ABC = 90 + x\) |
| M1 | full method to find the required angle eg a method leading to \(180 - x - x - 90\) | |
| A1 | for \(90 - 2x\) | |
| C2 | (dep M2) full reasons for their method, from base angles in an isosceles triangle are equal angles in a triangle add up to 180° a tangent to a circle is perpendicular to the radius (diameter) angles on a straight line equal 180° the exterior angle of a triangle is equal to the sum of the interior opposite angles | |
| (C1 | (dep M1) for a tangent to a circle is perpendicular to the radius (diameter) ) |
Additional guidance
Could be shown on the diagram alone
Needs to be an algebraic method
Accept \(x + x + 90 + y = 180\) for M2
Underlined words need to be shown; reasons need to be linked to their method; any reasons not linked do not credit.
C1: Apply the above criteria