Download Assignment unit 8 Geometry

Name:
Sha Tin College Mathematics Department
Key Stage 4 Extended Level Course
Unit 8 Assignment: Geometry
Need to Know
Angles on a line add to 180o
Total
/67(76 for JXL/CI)
Angle Relationships
Sketch the angle relationship here
Angles at a point add to360o
Vertically opposite angles are equal
Angles in a triangle add to 180o
Angles in an equilateral triangle are equal
ie. 60o
Base angles of isosceles triangles are equal
Corresponding angles in parallel lines are
equal
Alternate angles in parallel lines are equal
Co-interior angles in parallel lines add to
180o
Complementary angles add to 90o
Supplementary angles add to 180o
Sum of exterior angles of “n” sided
polygon is 360o
Exterior angle of regular “n”sided polygon
is 360o / n
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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Sum of interior angles of “n” sided polygon
is (n-2) x 180o
Interior angle of regular “n” sided polygon
is ((n-2)x 180o)/n
The angle in a semi-circle is 90o
The angle at the centre of a circle is double
the angle at the circumference.
Angles from the same arc are equal
Opposite angles in cyclic quadrilaterals are
supplementary.
The angle between a tangent and a radius
of a circle is 90o
Tangents from an external point are equal
in length.
Conditions that need to exist for triangles
to be congruent i.e. SAS, AAS, SSS and
RHS
If scale factor (SF) for length is k then
SF area = k2
SF volume = k3
A: Angle Relationships
#1 non calc In the diagram ABC is a straight line, angle EAB = 90º, angle BCD = 51º
and angle CDE = 125º. B is parallel to DC.
NOT TO SCALE
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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(a)
Calculate
(i)
(ii)
(b)
angle AED,
Answer (a)(i) angle AED =
[1]
Answer (a)(i) angle EBC =
[1]
angle EBC.
What is the special name of quadrilateral EDCB?
Answer (b) EDCB is a
[1]
Total for Section A
/3
B: Angles in Polygons
#1 non calc The hexagon ABCDEF has rotational symmetry of order 2 about O.
Angle FAB = 120º, angle ABC = 130º and angle CDE = 120º.
NOT TO SCALE
(a)
(b)
Write down angle DEF.
Answer (a) Angle DEF =
[1]
Answer (b) Angle BCD =
[2]
Calculate angle BCD.
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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#2
Each exterior angle of a regular polygon is 24º.
Calculate the number of sides of the polygon.
Answer
[2]
#3
NOT TO SCALE
Show by calculation, that an equilateral triangle, a regular polygon and
a regular 24 sided polygon fit together exactly at the point X, as shown
in the diagram.
Answer
[5]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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#4
#9
The diagram represents part of
a regular octagon ABCD . . .
with diagonal AC drawn.
NOT TO SCALE
(a)
(b)
Calculate angle ABC.
Answer (a) Angle ABC =
[2]
Answer (b) Angle ACD =
[2]
Calculate angle ACD.
Total for Section B
/14
C: Angles in Circles
#1
The chord AB of a circle, centre O, is parallel to the radius OT. Angle TAB = 41º.
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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Calculate
(a)
(b)
angle OTA,
Answer (a) Angle OTA =
[1]
Answer (b) Angle TOB =
[1]
angle TOB.
#2
AOB is a diameter of the circle,
centre O.
BC and OD are parallel.
ˆ  20 .
CBD
NOT TO SCALE
Find
(a)
(b)
(c)
(d)
ˆ ,
BDO
ˆ =
Answer (a) BDO
[1]
ˆ =
Answer (a) BDA
[1]
ˆ =
Answer (a) OAD
[1]
ˆ =
Answer (a) BCD
[1]
ˆ ,
BDA
ˆ ,
OAD
ˆ .
BCD
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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#3
AB is the diameter of a semicircle ACB.
The lines AG, CF and BE are parallel.
ˆ  y .
ˆ  x and CAG
EBC
NOT TO SCALE
(a)
(b)
ˆ .
Write down the value of ACB
[1]
ˆ =
Answer (b)(i) BCF
[1]
ˆ =
Answer (b)(ii) ACF
[1]
Write an expression for
(i)
(ii)
(c)
ˆ =
Answer (a) ACB
ˆ in terms of x ,
BCF
ˆ in terms of y .
ACF
Use your results from parts (a) and (b) to prove that x  y  90 .
Answer (c)
[1]
#4
The diagram represents a regular pentagon ABCDE inscribed in a circle, centre O.
The tangents at A and B meet at W.
NOT TO SCALE
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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Calculate
(a)
angle BCD,
[2]
(b)
angle CBD,
[2]
(c)
angle OAB,
[1]
(d)
angle WAB,
[1]
(e)
angle AWB.
[2]
Total for Section C
/18
D:Symmetry, Similarity
#1
Two different quadrilaterals each have one, and only one, line of symmetry.
In quadrilateral A, the line of symmetry is a diagonal.
In quadrilateral B, the line of symmetry is not a diagonal.
Draw each of the quadrilaterals, showing the line of symmetry, and write down
their special names.
Answer
QUADRILATERAL A
Name
QUADRILATERAL B
Name
[4]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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#2
Triangles P and Q are similar.
NOT TO SCALE
Their longest sides are 3 cm and 7 cm respectively.
(a)
Write down the ratio of their perimeters.
Answer (a) Perimeter of P : Perimeter of Q =
(b)
[1]
Calculate the ratio of their areas.
Answer (b) Area of P : Area of Q =
#3
:
:
[1]
The bowls shown in the diagram below are similar.
NOT TO SCALE
The capacity of the smaller bowl is 300 ml.
Calculate the capacity of the larger bowl.
Answer
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
ml [2]
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#4
The diagram shows a street light AE, which is 7 metres high.
A girl, who is 1.7 metres tall, stands 5 metres away from the point E.
Her shadow is x metres long.
NOT TO SCALE
Explain why
x
1.7

.
x5 7
Answer
#5
[1]
A, B and C are three similar containers.
Their heights are 40 cm, 30 cm and 15 cm respectively.
Container C has a surface area of 450 cm2 and has a capacity of 0.8 litres.
Calculate
(i)
the surface area of container A,
cm2 [3]
Answer (b)(i)
(ii)
the capacity of container B.
Answer (b)(ii)
litres [3]
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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#6
O is the centre of the circle.
Angle BOD = 132º.
The chords AD and BC meet at P.
(a)
(b)
(i)
Calculate angles BAD and BCD.
[2]
(ii)
Explain why triangles ABP and CDP are similar.
[1]
(iii)
AP = 6 cm, PD = 8 cm, CP = 3 cm and AB = 17.5 cm.
Calculate the lengths of PB and CD.
[4]
(iv)
If the area of triangle ABP is n cm2, write down, in terms of n,
the area of triangle CPD.
[2]
(i)
The tangents at B and D meet at T.
Calculate angle BTD.
[2]
Use OB = 9.5 cm to calculate the diameter of the circle
which passes through O, B, T and D, giving your answer
to the nearest centimetre.
[3]
(ii)
Total for Section D
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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Check List for Unit 8 Geometry
CAN DO STATEMENTS Unit 8 Geometry
Syllabus
Main Learning Objectives
Reference
4.1
RECAP (A) Know the meaning of these words with respect to
Geometry. Acute, obtuse, right angle, reflex, parallel,
perpendicular, equilateral, isosceles, regular, pentagon, hexagon,
octagon, rectangle, square, kite, parallelogram, trapezium,
Congruent
NEW Know the meaning of these words with respect to Geometry.
Similar, rhombus.
4.3
RECAP (A) Be able to measure and draw angles in degrees.
RECAP Be able to calculate missing angles by knowing the following angle properties.
4.4
Angles round a point add to 360o, angles on a straight line add to 180o, vertically opposite angles are
Tick here
equal, alternate angles on parallel lines are equal, corresponding angles on parallel lines are equal, cointerior angles on parallel lines are supplementary, angles in a triangle add to 180 o
4.4
4.9
4.9
4.8
4.2
4.6
4.6
4.6
RECAP Be able to calculate missing angles by knowing the
following angle properties. Angle sum of a triangle, quadrilateral
and polygons. Find interior and exterior angles of regular and
irregular polygons.
NEW Be able to calculate missing angles by knowing that;
the angle in a semi-circle is 90o, the angles at the centre of a circle
is double the angle at the circumference.
NEW Be able to calculate missing angles by knowing that; angles
from the same arc are equal and that opposite angles in cyclic
quadrilaterals are supplementary.
NEW Be able to calculate missing angles by knowing that the
angle between a tangent and a radius of a circle is 90o and tangents
from an external point are equal in length.
RECAP Be able to draw and describe the symmetry of a 2D and
3D shape. Including line and rotational symmetry.
NEW Understand the meaning of mathematical similarity. Use the
relationships between the areas and volumes of similar shapes to be
able to find missing dimensions.
NEW Find area and volume of similar figures using scale factor for
area and volume.
NEW Use the relationships between volumes and surface areas of
similar solids.
Sha Tin College Mathematics Department KS 4 ASSIGNMENT Geometry
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