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

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 ( ) (Interior angles regular figures) (  ’s at a point) Reflex  GEH = 240 ( ) (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

x x  AEB = 2  CBE x 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 x 2x Let  BED=x  BOD =2x (  at centre)

Read and interpret the information x 2x Let  OBD=90-x (base  isos. ∆) 90 - x

Read and interpret the information x 2x Let  DBC = x (rad  tang.) 90 - x x

Read and interpret the information x 2x  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  ∆) 2x x

x  EAB = 2x (base  ’s isos. ∆) 2x x

x  AEB = x (  sum ∆) 2x x 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

Now match reasons  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

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  )