November 2021 Paper 1 Q23
23 \(F\), \(H\), \(K\) and \(J\) are points on a circle.
Chords \(HJ\) and \(KF\) intersect at \(L\).
\(EFG\) is a tangent to the circle.
\(FH\) and \(JK\) are parallel.
Not drawn accurately
(a) Angle \(FHJ = 2x\)
Give reasons why angle \(FKJ\) and angle \(HJK\) are also equal to \(2x\). [2 marks]
(b) Work out the values of \(x\) and \(y\).
You must show your working.
Do not use trial and improvement. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Angles in the same segment | B1 | oe eg angles at the circumference are equal |
| Alternate angles | B1 | do not accept alternative or alternating |
Additional guidance
| Angles on the circumference from a chord | B1 |
| Angles in the same sector, opposite angles, parallel lines, angles from a chord, similar triangles, isosceles triangle, corresponding angles, triangles on a chord, intersecting chords, allied angles, alternate segment theorem | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(\angle HJF = 3y\) or \(\angle JFG = 2x\) or \(\angle HFL = 2x\) | M1 | may be on the diagram implied by one correct equation in \(x\) and \(y\) |
| \(2x + 3y + 98 = 180\) and \(4x + 7y = 180\) | M1dep | two correct equations in \(x\) and \(y\) |
| A correct attempt to eliminate one of the variables from the two equations | M1dep | eg \((4x + 7y) - 2(2x + 3y)\) |
| \(x = 17\) and \(y = 16\) | A1 |