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)