Higher November 2022 Paper 1 Q18
18
\(A\) and \(B\) are points on a circle, centre \(O\).
\(DBC\) is the tangent to the circle at \(B\).
Angle \(AOB = x^\circ\)
Show that angle \(ABC = \dfrac{1}{2}x^\circ\)
You must give a reason for each stage of your working. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| Result shown | M1 | for angle \(OBC = 90\) or for method to find angle \(OBA\) or angle \(OAB\), eg \(\dfrac{180 - x}{2}\) oe or for angle \(ABC = 90\) – angle \(OBA\), eg angle \(ABC = 90 - y\) or marks point on circumference and draws triangle using \(A\) and \(B\) and point marked |
| M1 | for method to find angle \(ABC\), eg \(90 - \text{``}\dfrac{180 - x}{2}\text{''}\) oe or for \(x = 180 - 2 \times\) angle \(OBA\), eg \(x = 180 - 2y\) or for angle at circumference \(= \dfrac{1}{2}x\) | |
| C1 | for correct algebra leading to angle \(ABC = \dfrac{1}{2}x\) and one circle theorem relevant to their method, eg The tangent to a circle is perpendicular to the radius OR for \(x = 180 - 2y\) and angle \(ABC = 90 - y\) and one circle theorem relevant to their method, eg The tangent to a circle is perpendicular to the radius OR for angle \(ABC = \dfrac{1}{2}x\) and one circle theorem relevant to their method, eg The angle at the centre of a circle is twice the angle at the circumference or Alternate segment theorem |
Additional guidance
Angles must be clearly labelled on the diagram or otherwise identified.
Correct method can be implied from angles on the diagram if no ambiguity or contradiction.
Underlined words need to be shown; reasons need to be linked to their method.