nzqa geometry excellence. sample 2001 read the detail line km forms an axis of symmetry. length qn =...
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NZQA Geometry
Excellence
Sample 2001
Read the detail
• Line KM forms an axis of symmetry.
• Length QN = Length QK.
• Angle NQM = 120°.
• Angle NMQ = 30°.
Read the detail
• Line KM forms an axis of symmetry.
• Symmetry is a reason• Length QN = Length
QK. • Isosceles triangle• Angle NQM = 120°.
• Angle NMQ = 30°.
Read the detail
• To prove KLMN is cyclic, you must prove that the opposite angles sum to 180 degrees.
Read the detail
QKN = 60• (Ext. isos ∆)60
Read the detail
QKN + QMN = 90
LKN + LMN = 180• (Symmetry)• Therefore KLMN is
cyclic.• (Opp. ’s sum to 180)
60
2002
2002
Read the information
• The logo is based on two regular pentagons and a regular hexagon.
• AB and AC are straight lines.
Interior angles in a hexagon
• Interior ’s sum to• (6-2) x 180 = 720
• Exterior angles in regular figures are
• 360/no. of sides.
• Interior angle is 180 minus the ext.
Interior angles in a hexagon
ADG = HFA = 360/5= 72
• (ext. regular pentagon)
DGE=EHF = 132(360-108-120)(Interior angles regular figures)(’s at a point)
Reflex GEH = 240(360-120)(Interior angles regular figures) (’s at a point)
Interior angles in a hexagon
Therefore DAF = 72(Sum interior angles of a
hexagon = 720)
2003
Read the information and absorb what this means
• The lines DE and FG are parallel.
• Coint ’s sum to 180
• AC bisects the angle DAB.
DAC=CAB
• BC bisects the angle FBA.
CBF=CBA
Let DAC= x and CFB= y
DAB = 2x
• (DAC=CAB)
FBA= 2y
• (FBC=CBA)
• 2x + 2y = 180
• (coint ’s // lines)
• X + y = 90
• I.e. CAB + CBA = 90
Let DAC= x and CFB= y
CAB + CBA = 90
• Therefore ACB = 90
• (sum ∆)
• Therefore AB is the diameter
• ( in a semi-circle)
2004
Read and interpret the information
• In the figure below AD is parallel to BC.
• Coint s sum to 180• Corr. s are equal• Alt. s are equal• A is the centre of the
arc BEF.• ∆ABE is isos• E is the centre of the
arc ADG.• ∆AED is isos
x
x
Let EBC = x
ADB = EBC = x
(alt. ’s // lines)
x
x
ADB = DAE = x
(base ’s isos ∆)
x
x
x
AEB = DAE + ADE = 2x
(ext. ∆)
x
2x
x
x
AEB = ABE
(base ’s isos. ∆)
x
2x
2x
x
x
AEB = 2CBE
x
2x
2x
= therefore
2005
Read and interpret the information
• The circle, centre O, has a tangent AC at point B.
• ∆BOD isos.• AB OB (rad tang)• The points E and D lie
on the circle. BOD=2 BED• ( at centre)
Read and interpret the information
x2x
Let BED=x
BOD =2x
( at centre)
Read and interpret the information
x2x
Let OBD=90-x
(base isos. ∆)
90 - x
Read and interpret the information
x2x
Let DBC = x
(rad tang.)
90 - x x
Read and interpret the information
x2x
CBD =BED = x
90 - x x
2006
Read and interpret
• In the above diagram, the points A, B, D and E lie on a circle.
• Angles same arc• Cyclic quad• AE = BE = BC.• AEB, EBC Isos ∆s• The lines BE and AD
intersect at F.• Angle DCB = x°.
x
BEC = x
(base ’s isos ∆)
x
EBA = 2x
(ext ∆)
2xx
x
EAB = 2x
(base ’s isos. ∆)
2xx
2x
x
AEB = 180 - 4x
( sum ∆)
2xx
2x
180-4x
2007
Question 3
• A, B and C are points on the circumference of the circle, centre O.
• AB is parallel to OC.• Angle CAO = 38°.• Calculate the size of angle ACB.
• You must give a geometric reason for each step leading to your answer.
Calculate the size of angle ACB.
Put in everything you know.
38
104
256
128
38
14
Now match reasons
38
104
256
128
38
14
ACO =38 (base ’s isos
AOC = 104 (angle sum )
AOC = 256 (’s at a pt)
ABC=128 ( at centre)
BAC=38 (alt ’s // lines)
ACB= 14 ( sum )
Question 2c
• Tony’s model bridge uses straight lines.• The diagram shows the side view of Tony’s model
bridge.
BCDE is an isosceles trapezium with CD parallel to BE.AC = 15 cm, BE = 12 cm, CD = 20 cm.
Calculate the length of DE.You must give a geometric reason for each
step leading to your answer.
Similar triangles
€
1220
=AB15
AB=9CB=6ED=6 isos trapezium
Question 2b
• Kim’s model bridge uses a circular arc.• The diagram shows the side view of Kim’s model
bridge.
WX = WY = UV = VX .UX = XY.
U, V, W and Y lie on the circumference of the circle.Angle VXW = 132°.
Calculate the size of angle WYZ.You must give a geometric reason for
each step leading to your answer.
Write in the angles and give reasons as you go.
WXY=48 (adj on a line)
Write in the angles and give reasons as you go.
WXY=48 (adj on a line)
XYZ=48 (base ’s isos )
Write in the angles and give reasons as you go.
WXY=48 (adj on a line)XYZ=48 (base ’s isos )
XWY=84 (sum )
Write in the angles and give reasons as you go.
WXY=48 (adj on a line)XYZ=48 (base ’s isos )
XWY=84 (sum )
WYZ=132 (ext)