GEOMETRY

1. In the diagram below ACE and BCD are straight lines.
i. Calculate the value of the angle marked x°
ii. Calculate the value of the angle marked y°.
iii. Calculate the value of the angle marked z°.
[Give two reasons for your answers.]
2. In the diagram below PQR is a right angled triangle. PQ is
parallel to TR.
PQ is 8.2 cm PT is 5.2 cm
TR is 3.6 cm TS is x cm
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i. What is the area of the trapezium PQRT?
ii. Find x which is the distance TS.
iii. Find tan θ.
3. In the triangle PQR
shown below, PQ =
QR and QR = PR.
i. What is the name
of this type of
triangle?
ii. What are the
values of the
angles marked x°
and y°?
4. Below is the plan of a farmer’s field.
What is the total area of the field?
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5. Use a ruler, a pencil, and a pair of compass only for this
question.
i. Draw a line segment, PQ, 7 cm long.
ii. Construct a line segment, LM, the perpendicular
bisector of PQ, such that LM cuts PQ at O, and
OL = OM = cm.
iii. Form a parallelogram PLQM by joining the points P, L,
Q and M.
iv. Measure and state the size of angle MPL.
v. What type of parallelogram is PLQM? Give a reason
for your answer.
6. The shape ABCD below is a trapezium.
The angle CAD is 19°.
The angle ACD is 98°.
BCE is a straight line.
i. What is the size of the angle marked with the letter p?
Give a reason for your answer.
ii. What is the size of the angle marked with the letter q?
Give a reason for your answer.
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7. Below is the diagram (not drawn to scale) of a regular
pentagon.
i. Calculate the value of
the angle marked q.
ii. Calculate the value of
the angle marked p.
8. Find the value of x in the diagram below.
9. Two angles are supplementary, if one of the angles is twice
the measure of the other, what is the measure of both
angles?
10. An exterior angle on a regular polygon measures 36°. How
many sides does the polygon have?
11. Each of the exterior angle of a regular polygon is 20°. How
many sides does the polygon have?
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12. The diagram below shows the quadrilateral PQRS.
NQ is a straight line parallel to PS.
i. What is the size of the angle marked a?
Give a reason for your answer.
ii. Work out the size of the angle marked b.
What property did you use to come up with your
solution?
13. Inside the regular hexagon below is
an equilateral triangle, a rhombus and
a trapezium.
a. What is the size of angle p?
b. What is the size of angle q?
c. The area of the equilateral
triangle is 5cm 2 . Work out the
total area of the regular hexagon.
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14. The figure below, not drawn to scale, is a regular hexagon
with centre X, and XY = 6 cm.
Calculate
i. the size of angle YZX
ii. the area of triangle YXZ, expressing your answer correct
to one decimal place.
iii. the area of the hexagon.
15. Two circles with centres P and Q and radii 5 cm and 2 cm
respectively are drawn so that they touch each other at T
and a straight line XY at S and R.
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16. ….
a. State with a reason
i. why PTQ is a straight line
ii. the length PQ
iii. why PS is parallel to QR
b. N is a point on PS such that QN is perpendicular to
PS.
Calculate
i. The length PN
ii. The length RS.
ABCDE is a regular pentagon inscribed in a circle centre O,
radius 12 cm, as shown in the diagram above, M is the
midpoint of DC.
i. Calculate the angle DOC (in degrees)
ii. Calculate DM
iii. Hence find the perimeter of the pentagon
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17.
In the diagram above (not drawn to scale) ABCDEF
represents a regular polygon. Given that the sum of the
interior angles of an n-sided polygon is 180(n – 2)°,
calculate the size of
i. angle AFE
ii. angle BCA
18. LMNOPQ is a hexagon(not drawn
to scale) with
P ˆ 110 , Qˆ
130 and Oˆ
90 ,
Lˆ Mˆ Nˆ
i. Calculate the value of .
ii. Given that PO = 4cm and the
area of ΔNOP = 12 cm 2 .
iii. Calculate the length of PN in
cm, giving your answer correct
to one decimal place.
ˆL
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19. ….
ABCDE is a pentagon, not drawn to scale, with
Aˆ Bˆ Dˆ x.
Angle C = 72° and angle E = 81°.
Calculate the value of x.
20. The angles of a quadrilateral taken in order are 90°, x°, 2x°,
3x°.
i. Calculate the size of the unknown angles.
ii. Name the type of quadrilateral.
21. The diagram, not drawn to scale, represents a regular
polygon PQRST.
i. Given that the sum of the
interior angles of the
polygon with n sides is
180(n – 2)°, calculate angle
TSR and angle TRS.
ii. Given further that angle
PQR = x°, and angle
RTP = y°, show that
y
2. x
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22.….
In the diagram above, not drawn to scale, AB is parallel to
CD and EG is parallel to FH. Angle IJL = 50° and angle
KIJ = 95°
Calculate the values of x, y, and z showing clearly the steps
in your calculations.
23. In the diagram above, not
drawn to scale, BG is
parallel to DE, AF and CH
are straight lines. Calculate
the values of x and y
showing clearly the steps
in your calculation.
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24.
ABCDE is a pentagon. The angles A, B, C, D, and E are x°,
2x°, 3x°, 4x°, and 120°. Find the value of x.
25. In the diagram, not
drawn to scale, angles
BAC = 24°, EDC = 30°
and angle CED = x°.
Calculate in terms of x
the size of
i. Angle AFD
ii. Angle BFE
iii. Hence deduce the
value of x.
26. Given that π radians = 180° express
i.
radians in degrees
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ii. 210° in degrees
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27. In the diagram, ABC and ACD are isosceles triangles in
which AB = AC = AD and
BAC
ACD
74
a. Calculate
i.
ii.
b. What special type of quadrilateral is ABCD?
CAD
BCA
28. State three properties which define a rhombus, with respect
to its sides, angles and diagonals
ABCD is a rhombus (not drawn to scale) with AO = 4.8cm
and BC = 3.6 cm
Calculate
i. The length of AB.
ii. The state of angle BAD to the nearest degree.
iii. The area of ABCD.
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29.
The minute hand of a clock is 25 cm long and the hour
hand is 21 cm long. Calculate
i. the distance moved by the tip of the hour hand when
the time goes from 2 a.m. to 10 a.m. the same day.
ii. the smaller angle between the hands of the clock when
the time is 10 o’clock.
30. In this question, use 3.14
A wheel is turning at a rate of 33 revolutions per minute.
Express this speed in radians per second, correct to 2
significant figures
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31. …
Similar Triangles
32. ….
In the diagram above, not drawn to scale, PQ, XY, and SR
are parallel lines. QY = 10cm, YR = 5cm and XY = 3cm.
i. Prove that triangle PQX and RSX are similar.
ii. Calculate the lengths of PQ and RS.
iii. Calculate the ratio of the areas of the triangles PQX
and RSX.
BAC
80
In the figure above, not drawn to scale,
AC, MB = MC and BF is perpendicular to AC.
i. Calculate the angles ABF, BMC, and MCF.
ii. Show that triangles ABF and MCF are similar.
. AB =
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33.
Triangles PXY and PQR, not drawn to scale, are similar
triangles. PX = 4 cm, XQ = 6 cm, and
QR = 8 cm. Calculate
i. The length of XY in cm.
ii. The area of triangle PXY given that the area of triangle
PQR is 50 cm 2 .
34.
The figure SJKM above, not drawn to scale, is a trapezium
with SJ parallel to MK, angle MJK = 124°, image
MSJ = 136° and SM = SJ = 50 m.
a. Calculate the size of
i. Angle SJM
ii. Angle JKM
iii. Calculate, expressing your answer correct to one
decimal place, the length of MJ.
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