circle theorems standard questions (g10) · circle theorems standard questions (g10) the oakwood...
TRANSCRIPT
The Oakwood Academy
Page 1
Circle Theorems Standard Questions (G10)
The Oakwood Academy
Page 2
Q1.(a) A, B and C are points on the circumference of a circle with centre O.
Not drawn accurately
Work out the size of angle x.
Answer ........................................................ degrees (1)
The Oakwood Academy
Page 3
(b) P, Q and R are points on the circumference of a circle with centre O.
Not drawn accurately
Work out the size of angle y.
Give a reason for your answer.
Answer ........................................................ degrees
Reason ...........................................................................................................
......................................................................................................................... (2)
(Total 3 marks)
The Oakwood Academy
Page 4
Q2.AB is a tangent to the circle.
Not drawn accurately
Write down the value of x. Give a reason for your answer.
Answer ........................................................ degrees
Reason ...................................................................................................................... (Total 2 marks)
The Oakwood Academy
Page 5
Q3. (a) AB is a chord of the circle.
TA and TB are tangents to the circle.
Not drawn accurately
Work out the size of angle x.
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
Answer ........................................................ degrees (2)
The Oakwood Academy
Page 6
(b) The diagram shows a circle, centre O.
C and D are points on the circumference. COE is a straight line. DE is a tangent.
Not drawn accurately
Work out the size of angle y.
You must show your working, which may be on the diagram.
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
Answer ........................................................ degrees (3)
(Total 5 marks)
The Oakwood Academy
Page 7
Q4.The diagram shows a circle, centre O.
Not drawn accurately
(a) Work out the size of angle x.
.........................................................................................................................
Answer ........................................................ degrees (1)
(b) Work out the size of angle y.
.........................................................................................................................
.........................................................................................................................
Answer ........................................................ degrees (3)
(Total 4 marks)
The Oakwood Academy
Page 8
Q5. Points A, B, C and D are on the circumference of the circle. CDE is a straight line.
Not drawn accurately
(a) Work out the size of angle BCD. Give a reason for your answer.
Answer ........................................................ degrees
Reason .......................................................................................................... (2)
(b) Work out the size of angle ABC.
..........................................................................................................................
Answer ........................................................ degrees (1)
(Total 3 marks)
The Oakwood Academy
Page 9
Q6. PAQ and PB are tangents to the circle.
AC = BC
Not drawn accurately
Work out the size of angle x.
You must show your working which may be on the diagram.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
Answer ........................................................ degrees (Total 4 marks)
The Oakwood Academy
Page 10
Q7.(a) P, Q, R and S are points on the circumference of the circle.
Not drawn accurately
Work out the size of angle x.
..........................................................................................................................
Answer ...................................................................... degrees (1)
(b) The diagram shows a circle and a tangent at T.
Not drawn accurately
Write down the size of angle w.
Give a reason for your answer.
Answer ...................................................................... degrees
Reason ............................................................................................................ (2)
The Oakwood Academy
Page 11
(c) A, C and D are points on the circle, centre O.
BA and BC are tangents to the circle.
Not drawn accurately
Work out the size of the angle y.
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
Answer ...................................................................... degrees (3)
(Total 6 marks)
The Oakwood Academy
Page 12
Q8.
(a) O is the centre of the circle.
Write down the size of angle a.
Not drawn accurately
Answer ........................................................ degrees (1)
(b) Write down the size of angle b.
Give a reason for your answer.
Not drawn accurately
Answer ........................................................ degrees
Reason ..........................................................................................................
........................................................................................................................ (2)
The Oakwood Academy
Page 13
(c) A, B, C and D are points on a circle, centre O.
Not drawn accurately
x + y + z = 290°
Work out the values of x, y and z.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
x = ............................................................... degrees
y = ............................................................... degrees
z = ............................................................... degrees (3)
(Total 6 marks)
The Oakwood Academy
Page 14
Q9.(a) The diagram shows a circle, centre O.
Not drawn accurately
Work out the value of x.
.........................................................................................................................
Answer ........................................................ degrees (1)
(b) The diagram shows a cyclic quadrilateral.
Not drawn accurately
Work out the value of y.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
Answer ........................................................ degrees (2)
(Total 3 marks)
The Oakwood Academy
Page 15
Q10.A, B, C, D and E are points on a circle.
AC is parallel to ED.
Not drawn accurately
(a) Write down the size of angle x.
Answer ........................................................ degrees (1)
(b) Work out the size of angle y.
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
Answer ........................................................ degrees (3)
(Total 4 marks)
The Oakwood Academy
Page 16
Q11. P, Q and R are points on a circle. Triangle PQR is isosceles. XY is a tangent to the circle at P.
Not drawn accurately
Work out the size of angle RPY.
..................................................................................................................................
..................................................................................................................................
..................................................................................................................................
..................................................................................................................................
Answer ....................................................... degrees (Total 2 marks)
The Oakwood Academy
Page 17
Q12. A, B, C and D are points on the circumference of a circle centre O. AC is a diameter. Angle DAC = 42°
Not drawn accurately
(a) Write down the value of x.
Answer ...................................................................... degrees (1)
(b) Work out the value of y.
Answer ...................................................................... degrees (1)
(Total 2 marks)
The Oakwood Academy
Page 18
M1.(a) 35 B1
(b) 100 B1
Angle at centre twice angle on circumference
Must use words ‘centre’ and ‘circumference’ (or ‘perimeter’)
Allow poor spelling even though both words given
oe (strand) (i) Q1
[3]
M2.43 B1
Alternate segment (theorem)
Strand (i) Do not accept Alternate
Dependent on B1 Q1
[2]
M3. (a) (180 − 38) ÷ 2
M1
71
May be on diagram if no contradiction A1
(b) ODE = 90 seen or implied or CDO = 25 or COD = 130
The Oakwood Academy
Page 19
May be on diagram B1
DOE = 50 or CDE = 115 or 140 seen
May be on diagram M1
40 A1
Additional Guidance 40 with no working seen
B1M1A1
115 is B1M1 unless from clearly incorrect working e.g. 115 leading to an answer of 65 is M1 only
[5]
M4.(a) 150 B1
(b) 360 − 150 or 210
or 360 − their 150
oe
OCA = 18 seen or implied
or 180 − 18 − 75 or 87 M1
360 − 18 − 75 − 210
or 360 − 18 − 75 − their 210
oe
OCB = 75 − 18 or 57 seen or implied
180 − 93 − 30 or 87 − 30 M1dep
57 A1
[4]
The Oakwood Academy
Page 20
M5. (a) 108
B1
Opposite angle of a cyclic quadrilateral
Strand (i)
(add up to 180)
Must have 108 Q1
Additional Guidance Must see “opposite” and “cyclic” (oe e.g. quadrilateral in a circle)
(b) 125 B1
[3]
M6. ABC = 52
May be on diagram M1
BAC = 52
or BAQ = 104
or ACB = 76
May be on diagram M1dep
PAB = 76
or PBA = 76
May be on diagram M1dep
28
Clear evidence that 28 is for angle x A1
The Oakwood Academy
Page 21
Additional Guidance Angles may be on diagram ACB = 52 and ABC = 52
M1 only
[4]
M7.(a) 56 B1
(b) 70 B1
Alternate segment (theorem)
Strand (i)
Dependent on B1 Q1dep
(c) 2 × 47 or 94
or Angle BOA = 47
or Angle BOC = 47
or Angle BAC = 47
or Angle BCA = 47
May be on diagram (obtuse angle) M1
90 or right angle symbol seen at A or
C
or 180 − 90 − 47
or (180 − 2 × 47) ÷ 2
oe M1
The Oakwood Academy
Page 22
43 A1
[6]
M8. (a) 35
B1
(b) 40 B1
Opposite angles of cyclic quad (add up to) 180°
oe
Strand (i) No need to mention 180 if angle given as 40
Accept ‘supplementary’ to mean adds to 180. Q1
(c) x = 55
y = 110
z = 125
If answer line blank mark diagram or script.
B2 any two correct
B2 y = 110 and x + z = 180
B2 z = 125 and y = 2x
B1 any one correct
B1 values less than 180 such that y = 2x or x + z = 180 B3
[6]
M9.(a) 70 B1
(b) 4y + y = 180 or 5y = 180 oe
or 180 ÷ 5
The Oakwood Academy
Page 23
M1
36 A1
[3]
M10.(a) 70 B1
(b) ADE = 34
or AED = 180 − 70 or 110
or ADC = 180 − 70 − 34 or 76
Angles seen on diagram must be in correct place M1
ADE = 34
and AED = 180 − 70 or 110 M1dep
36 A1
[4]
M11. 180 − (2 × 35)
RQP = 110 marked on diagram B1
110 B1
[2]
The Oakwood Academy
Page 24
M12. (a) 42
B1
(b) 48 B1
[2]