MathedUpCircle TheoremsRemix Worksheet

Exam-style questions written to the standard of 102 published papers. Each is labelled with the board and tier it is pitched at.
This sheet: —

Circle theorems

Exam questions, remixed

Answer every question. Show your working: marks are given for method. The board and tier each question is pitched at is printed above it.

1Highercalculator · about 3 min
(a)
PP, QQ and RR lie on a circle.
Point SS sits inside triangle PQRPQR.
150°xQPRSNot drawn accurately
Assume that SS is the centre of this circle.
Work out the size of angle xx.
x = °[1]
(b)
In fact the centre of the circle lies on PSPS but not at SS.
What does this mean for angle xx?
Tick one box.
The same as your part (a) answer
Greater than your part (a) answer
Smaller than your part (a) answer
Impossible to tell
[1]
(c)
For a different circle,
   ABAB is a tangent to the circle at AA
   CC and DD lie on the circumference
   AC=CDAC = CD
78°yACDBNot drawn accurately
Simon writes out the method below to find the size of angle yy.
Angle ADC=78∘ADC = 78^\circ  (alternate segment theorem)
Therefore y=78∘y = 78^\circ  (angles in an isosceles triangle)
Is he right?
Give a reason for your answer.
[1]
2Highernon-calculator · AO1 · about 3 min
The diagram shows a cyclic quadrilateral.
  a:b:c = 8:19:12a : b : c \ = \ 8 : 19 : 12
cdabNot drawnaccurately
d = °[3]
3Highercalculator · AO1 · about 3 min
PQPQ is a diameter of a circle.
QRQR is a tangent to the circle.
PQ=QRPQ = QR
PR=16PR = 16 cm
PQRNot drawnaccurately
Work out the radius of the circle.
Give your answer as a decimal.
cm[3]
4Highercalculator · AO2 · about 5 minSYNOPTIC
In the circle shown, EE is the centre and ACAC passes through it as a diameter.
EE also bisects the chord BDBD, so EE is the midpoint of BDBD.
ACBDENot drawnaccurately
Prove that triangles ABEABE and CDECDE are congruent.
[4]
5Highernon-calculator · about 6 minSYNOPTIC
TT, UU and VV are points on a circle.
VWVW is a tangent to the circle.
TUVWNot drawn accurately
Assume triangle TUVTUV is isosceles, with TV=UVTV = UV
(a)
Prove that TUTU is parallel to WVWV.
[4]
In fact, the triangle TUVTUV is equilateral.
(b)
Tick the two statements that must be correct.
TVTV bisects angle UTWUTW
TUTU and WVWV are parallel
TVTV bisects angle UVWUVW
[1]
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