June 2022 Paper 1 Q15
15 \(B\), \(C\) and \(D\) are points on a circle, centre \(P\).
\(AB\) and \(AC\) are tangents to the circle.
Not drawn accurately
Prove that \(\quad y = 90 + \dfrac{x}{2}\) [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| States that \(\angle ABP\) or \(\angle ACP\) is 90 | B1 | can be seen on diagram (either 90 or a square angle) |
| Any one further angle correct (not \(\angle ABP\) or \(\angle ACP\)) | B1 | minor \(\angle BPC = 180 - x\) or \(360 - 2y\) or major \(\angle BPC = 2y\) or \(180 + x\) or \(\angle BQC = 180 - y\) or \(90 - \dfrac{x}{2}\) (where Q is a point on the major arc) |
| Another further angle correct (not \(\angle ABP\) or \(\angle ACP\)) | B1 | any two of minor \(\angle BPC = 180 - x\) or \(360 - 2y\) or major \(\angle BPC = 2y\) or \(180 + x\) or \(\angle BQC = 180 - y\) or \(90 - \dfrac{x}{2}\) (where Q is a point on the major arc) could be the same angle found in the previous B mark but an expression in \(y\) rather than \(x\) |
| A correct equation in terms of \(x\) and \(y\) and rearrange to \(y = 90 + \dfrac{x}{2}\) | B1dep | dependent on first three B marks awarded doesn’t imply the first 3 B marks |
| 3 reasons given for the theorems used correctly for the angles stated in the first three marks | B1dep | dependent on first three B marks awarded reason - angle formed from a tangent and a radius is a right angle (can only be used once) reason - angles in a quadrilateral add up to 360 reason - angle at the centre is twice the angle at the circumference reason - opposite angles in a cyclic quadrilateral add up to 180 reason - angles at a point (or in a circle) add up to 360 reason - alternate segment theorem |
Additional guidance
| Angles must be identified with either our terminology such as \(\angle ABP\) or their own labelling such as m or \(\theta\) or can be seen on the diagram Accept supplementary for angles adding to 180 Accept complementary for angles adding to 90 Use of obtuse and reflex or interior and exterior instead of minor and major is fine. If it’s not clear then assume it’s the minor arc they are referring to Check candidates are not assuming that \(BDCP\) is a kite and using symmetry of this shape Check candidates are not using \(BDCP\) as a cyclic quadrilateral No credit for numbers used instead of \(x\) and \(y\) Mark the first three B marks positively Note – \(ABPC\) is a cyclic quadrilateral but \(D\) is not the centre of that circle Note – \(D\) is not the middle of minor arc \(BC\) |