Maths Level 4 Module 9

Mathematics NQF Level 4 *see terms and conditions Downloaded from hlayiso.com Mathematics NQF Level 4

Space, shape and measurement Topic 3 Downloaded from hlayiso.com Mathematics NQF Level 4

Explore, interpret and justify geometric relationships Module 9 Downloaded from hlayiso.com Mathematics NQF Level 4

Learning activity 9.1 Downloaded from hlayiso.com Mathematics NQF Level 4

2. From Figure 9.12, name: 2.1) Supplementary angles ˆ + COB AOC ˆ ˆ + BOD COB ˆ ˆ + DOA BOD ˆ ˆ + AOC DOA ˆ Downloaded from hlayiso.com Learning activity 9.1

2. From Figure 9.12, name: 2.2) An obtuse angle ˆ BOC ˆ AOD 2.3) An acute angle ˆ BOD ˆ AOC Downloaded from hlayiso.com Learning activity 9.1

2. From Figure 9.12, name: 2.4) One pair of vertically opposite angles ˆ and COA BOD ˆ or ˆ and COB AOD ˆ 2.5) A reflex angle ˆ BOD ˆ COB or ˆ AOC ˆ DOA Downloaded from hlayiso.com Learning activity 9.1

Learning activity 9.3 Downloaded from hlayiso.com Mathematics NQF Level 4

a)4. BÂCABCE is a cyclic quadrilateral. Determine the b)magnitude AĈE of: [VIDEO] Click above to play this video. Downloaded from hlayiso.com Learning activity 9.3

Learning activity 9.4 Downloaded from hlayiso.com Mathematics NQF Level 4

ˆ = 15°, ABO 4. If determine the: a) magnitude of b) magnitude of c) magnitude of d)length of AP e) length of BP. [VIDEO] Click above to play this video. Downloaded from hlayiso.com Learning activity 9.4

Summative assessment Module 9 Downloaded from hlayiso.com Mathematics NQF Level 4

1. Determine the angles of all the triangles in the diagram below: Figure 9.15 Downloaded from hlayiso.com Summative assessment – Module 9

1. Continued ˆ = 60° • ACB [∆ ABC is an isosceles ∆] ˆ = 180° − 120° [interior angles of a ∆] • ABC = 60° ˆ = 180° − 60° [straight angle] • BCD = 120° Figure 9.15 60˚ 60˚ 120˚ Downloaded from hlayiso.com Summative assessment – Module 9

1. Continued ˆ • CBD D = [isosceles ∆] ˆ = 180° [interior angles of a ∆] ˆ + BCD 2CBD ˆ = 180° -120° 2CBD ˆ ∴ CBD Dˆ 30 ==° Figure 9.15 60˚ 30˚ 60˚ 120˚ 30˚ Downloaded from hlayiso.com Summative assessment – Module 9

2. AE  BD; FD  AC and FA  DC Determine the magnitude of: a) FÊA b) BDC ˆ Figure 9.16 Downloaded from hlayiso.com Summative assessment – Module 9

2. a) Determine FÊA ˆ = 40° EAB [corresponding angles] ˆ ˆ = EAB FEA [alternating angles] = 40° b) Determine BD̂C ˆ = DBC EDB ˆ 40˚ 40˚ = 40° 40˚ 40˚ [alternating angles] Downloaded from hlayiso.com Figure 9.16 Summative assessment – Module 9

2. b) Determine BD̂C Continued ˆ = 180° − 60° GFA [straight angle] = 120° ˆ = EDC [corresponding angles] ˆ + BDE BDC ˆ = EDC ˆ 120˚ 40˚ 40˚ 120˚ ˆ + 40° = 120° BDC 80˚ ˆ = 80° ∴ BDC 40˚ 40˚ Downloaded from hlayiso.com Figure 9.16 Summative assessment – Module 9

3. Use the diagram to calculate the magnitude of: a) DAB ˆ ˆ b) ABC ˆ c) DCB Figure 9.17 Downloaded from hlayiso.com Summative assessment – Module 9

3. ˆ = BCE 1) DAB ˆ [Exterior angle of a cyclic quad] = 85° 2) B̂= 90° [∠ at centre = 2 x ∠ on circle] 3) DCBˆ = 180° − DAB ˆ 85˚ = 180° − 85° = 95° 95˚ [straight angle] Figure 9.17 Downloaded from hlayiso.com Summative assessment – Module 9

4. If OC = 50 cm and arc AC = 60 cm, calculate the magnitude of angle B. ˆ =θ Let AOC arc length NOTE: θ is θ= in radians radius here 60 = rad 68.76˚ 50 = 1, 2 rad =1, 2 × 57,3 = ° 68, 76° Figure 9.18 180 [Degrees = radians × ] Downloaded from hlayiso.com π Summative assessment – Module 9

4. If OC = 50 cm and arc AC = 60 cm, calculate the magnitude of angle B. ˆ AOC = 68, 76° 34,38˚ 1 ˆ ∴ B̂ = AOC 2 1 68,76˚ = (68, 76°) 2 = 34,38° Figure 9.18 [∠ at centre = 2 x ∠ on circle] Downloaded from hlayiso.com Summative assessment – Module 9

ˆ 5. Calculate the magnitude of ATB = 180° − 50° − 60° Ĉ [interior angles of ∆] = 70° ˆ = ATB C ˆ 70˚ = 70° [tan-chord theorem] 70˚ Figure 9.67 Downloaded from hlayiso.com Summative assessment – Module 9

6. In the figure given below, MN  OP and VQ = NQ . NVQˆ = 63° . Determine, giving reasons, the values of a, b, c, d, e and f: Figure 9.68 Downloaded from hlayiso.com Summative assessment – Module 9

6. Continued • aˆ + 63= ° 180° [Adjacent supplementary angles] ˆ 180° − 63° a = = 117° • bˆ = NVQ ˆ == 63° [VQ NQ; Isosceles ∆VQN ] ˆ + bˆ + cˆ= 180° • NVQ [Interior angles of ∆VQN are ˆ − bˆ ˆ 180° − NVQ c = supplementary] = 180° − 63° − 63° = 54° Downloaded from hlayiso.com Summative assessment – Module 9

6. Continued • dˆ =cˆ =54° [Vertically opposite angles] • eˆ =bˆ =63° [MN  OP; Alternate angles are equal] • fˆ + 63= ° 180° [MN  OP; Co-interior ˆ 180° − 63° f= angles are equal] = 117° Downloaded from hlayiso.com Summative assessment – Module 9

7. In the diagram below, O is the centre of the circle with diameters AC and BD. AB  DC and AC meets BD at O. P is any point on the minor arc BC, PB and PC are joined. Calculate , with reasons, the measures of angles labelled x, y and z. Downloaded from hlayiso.com Figure 9.78 Summative assessment – Module 9

7. Continued ˆ = 130° • xˆ + B [ext ∠ of triangle] 1 ˆ xˆ = B [AO = BO, radii] 1 ∴ 2 xˆ = 130° ∴ xˆ = 65° • yˆ =xˆ =65° [ AB  DC ; Alternate angles are equal] Figure 9.78 Downloaded from hlayiso.com Summative assessment – Module 9

7. Continued ˆ = 360° -130° • O [Angles around Point O add up to 360°] = 230° Ô 230 ° zˆ = = 2 2 = 115° [∠ at centre = 2 x ∠ on circle] Downloaded from hlayiso.com Figure 9.78 Summative assessment – Module 9

8. Refer to the diagram below. AB is a tangent to the circle with centre O. BCOD is a straight line. AE  BD and D̂ = x . Find, with reasons, five other angles each equal to x . Downloaded from Figure 9.70 hlayiso.com Summative assessment – Module 9

8. Continued Â1 = x [AE  BD ; Alternate angles are equal] Ê1 = x [Angles subtended by chord AC are equal] ˆ= Eˆ= x [AE  BD ; Alternate angles are equal] C1 1 Ê= ˆ= x C [OE = OC; radii are equal ; Isosceles ∆OEC] 2 1 ˆ= D A ˆ= x 3 Downloaded from hlayiso.com Summative assessment – Module 9

Mathematics NQF Level 2 Downloaded from hlayiso.com Mathematics NQF Level 4