Maths-Investigation-Grade-10-March-2024-QP-and-Memo_hlayiso.com_.pdf

Downloaded from Stanmorephysics.com GRADE 10 FEBRUARY 2024 MATHEMATICS TIME: 60 MIN INVESTIGATION TOTAL: 50 NAME: 2... ccc ec eee nett nee e eens INSTRUCTIONS: > You may use a calculator, round your answers off to one decimal place where needed. > Write neatly and legible > Read through all the hints and answer all the questions Question _| Question Total Learners total per uestion 1 6 2 13 3 10 4 16 5 5 Total 50 Page 1 of 9

Downloaded from Stanmorephysics.com COMPLETE THE FOLLOWING: In a right-angled triangle, we can name the sides of the triangle according to the position of the angle and the right angle: Adjacent side" Or Opposite side Adjacent side Investigate the different RATIOS of all the sides in similar triangles. The four triangles below are given from smallest to biggest. The corresponding sides opposite the common angle, @, is shown on the sketch as parallel lines. The angles which are equal is shown as *. USE THE FOLLOWING SKETCH TO ANSWER ALL THE QUESTIONS IN THIS INVESTIGATION. J Page 2 of 9

Downloaded from Stanmorephysics.com Question 1 [6] Given: _ the lengths of the lines: AB= 49cm, AD= 85cm, AF= 11,8cm and AH=13,9 cm AC= 54cm, AE=9,4cm, AG= 13cm and AJ= 15,3 cm CB= 23cm, ED= 4cm, GF=5,5cm and JH=6,4 cm 1.1. Complete the similar triangles by writing the letters of the triangles in the correct order. AABC ||| A Il A I| 4 [3] 1.2. Whyare the four triangles similar to each other? [1] 1.3. Why is it important to write the letters in the correct order when it comes to similarity? [2] uestion 2 [13 2.1 Complete the table by writing down THE CORRECT SIDES of the triangles: AADE, AAFG and AAHJ [3] Name of the side AABC AADE AAFG AAHJ] Opposite side (O) CB Adjacent side (A) AB Hypotenuse (H) AC 2.2 Complete the table below: [4] Name of the side AABC AADE AAFG AAHJ Opposite side (O) cm cm cm cm Adjacent side (A) cm cm cm cm Hypotenuse (H) cm cm cm cm Page 3 of 9

Downloaded from Stanmorephysics.com 2.3 Complete the RATIOS by using the previous two tables and use your calculator to round it off to one decimal place [3] Name of side AABC AADE AAFG AAHJ o ite sid pposite side aa) __ = on — ma fe a hypotenus adjacent side a —o cm —s cm — Ss cm hypotenuse opposite side~ | a —Ee cm | —= cm | —= cm adjacent side 2.4 Check the values of the ratios for every triangle like we calculated previously. . . ite sid 2.4.1 Did the value of the ratio of “PP°** °° change or stayed the same for all the hypotenuse different triangles? [1] 7 2 dj id. 2.4.2 Did the value of the ratio of jae oat L change or stayed the same for all the hypotenuse different triangles? [1] ‘ P ite sid 2.4.3 Did the value of the ratio of pl Ee change or stayed the same for all the adjacent side different triangles? [1] Page 4 of 9

Downloaded from Stanmorephysics.com uestion 3 [10 3.1. Ifthe size of angle @ in the sketch on page 3, is: @ = 25° 3.2 Use your calculator to determine the value of the following to ONE decimal place: sin25°= cos25° = tan25° = [3] 3.3. Compare your answers in table 3.2 and table 2.3: ( choose only one option by marking the corrrect column on the righthand side by X ) e sin25° had the same answer in 2.3 and 3.2 as: (choose only one option by marking the corrrect column on the righthand side by an X ) [3] opposite side adjacent side opposite side hypotenuse hypotenuse adjacent side ¢ cos25° had the same answer in 2.3 and 3.2 as: (choose only one option) opposite side adjacent side opposite side hypotenuse hypotenuse adjacent side e tan25° had the same answer in 2.3 and 3.2 as: (choose only one option) opposite side adjacent side opposite side hypotenuse hypotenuse adjacent side CONCLUSION: 3.4 Do you think that the previous RATIOS wil always be the same for THE SAME ANGLE SIZES although the triangles may differ in size? Explain your answer. [3] Page 5 of 9

Downloaded from Stanmorephysics.com Question 4 [16] In the following excercise the right angled AXYZ is given: Z 4.1 Inthe given triangle there are three sides. Write down which one of the sides XY, YZ or XZ will be the: [3] e opposite side of 35,3°: e adjacent side of of 35,3°: e hypotenuse of the rightangled triangle: 4.2 If YZ is the WANTED SIDE and XZ is the GIVEN SIDE, which ratio of 2 = sin35,3° or < = cos 35,3° or 2 = tan 35,3° will you use to determine YZ? [3] 4.3 Use 4.2 to determine YZ, if XZ = 32cm [5] Page 6 of 9

Downloaded from Stanmorephysics.com 4.4 Determine the length of XY in the same way you just did by choosing the correct ratio: [5] Question 5 [5] Angie Of Elevation: Nis the angle between a horizontal line tree ty 50m B 5.1 Ryan (R) is looking up at the cliff of a mountain. He is 50 m from B, the bottom of the mountain and the angle of elevation from where he stands is 66°. Determine the perpendicular height of the mountain. (Use question 4 as a guideline. The diagram is not drawn according to scale) Page 7 of 9

Downloaded from Stanmorephysics.com TOTAL 50 Page 8 of 9

Downloaded from Stanmorephysics.com SIMILAR TRIANGLES Triangles are similar if the: - corresponding angles are equal. (Then the corresponding sides are in the same ratio.) OR - the corresponding sides are in the same ratio. (Then the corresponding angles are equal ) A From the 2 triangles In AABC and APQR A=P_ given B=Q _ given C=R_ given AABC ||| APQR [AAA] (three corresponding angles of the two triangles are =) AB = PQ and AC = PR and BC = QR Page 9 of 9

Downloaded from Stanmorephysics.com No Calculations Mark Mar | Level Allocatin k Question 1 [6] 1.1 Complete the similar triangles through writing the letters of v AADE FE) 2 the triangles in the correct order: v AAFG v DAH ¥ AADE v AAFG v AAW) 1,2 Why are the four triangles similar to each other? Y any valid 1 1 reason Y all the triangles are equiangular / corresponding angles are equal/ The triangles have the same shape 1.3 Why is it important to write the letters in the correct order ¥ to determine 2 2 when it comes to similarity? the corresponding sides Y to determine the corresponding sides Y and corresponding angles Y and corresponding angles [6] Question 2 [13] 24 ¥ Column «ADE | 3 1 ¥ Column « AFG " Y¥ Column « AHJ Name of the . , t | AFG | AAHJ GF JH AF AH AG AJ v v 2.2 ¥ Column «ABC | 4 1 Name ofthe | AABC | AADE AAFG AAHJ side (0) CB ED GF JH =2,3cm =4cm | =5,5cm | = 64cm (A) AB AD AF AH =4,9cm = 85cm} = 11,8cm| = 13,9cm (H) AC AE AG AJ =5cm =9,4cm| = 13cm | = 15,3cm ¥ Column « ADE ¥ Column « AFG ¥ Column « AHJ

Downloaded from Stanmorephysics.com Name 2.3 ofthe | AABC | AADE | AAFG | AAHJ oO 3 side ¥ row — h Oo 2,3 4 5,5 6,4 = 54 9,4 13 15,3 ho | =04 | =04 | =04 | =04 a Y row — 4,9 h aa 8,5 11,8 13,9 a 54 9,4 13 15,3 0 h =09 | 2o9 | =09 | =09 v row — a 2,3 4 55 | 64 oO 49 —_— — —— = _ 05 8,5 11,8 13,9 a , =0,5 =05 | =0,5 2.4.1 | ¥ The same v The same 1 2.4.2 | ¥ The same v The same 1 2.4.3 | ¥ The same v The same 1 [13] Question 3 [10] 3.1 ¥ 0 =25° 1 3.2 v sin25° = 0,4 3 Y¥ cos 25° = 0,9 v tan 25° = 0,5 . : t 33° |v sin 25° =~ 3 Ss a ¥ cos 25° = 5 t Y tan25° = : ¥ tan25° = z 3.4 vyes v yes 3 vit must be a right angled triangle Y right angled v there must be another angle which is also equal triangle Y another angle equal [10]

Downloaded from Stanmorephysics.com Question 4 [16] 4.1 Y opposite side to 35,3°: ZY Y opposite side to | 3 Y adjacent side to 35,3°: XY 35,3°: XY v hypotenuse of the right-angled triangle: XZ Y adjacent side to 35,3°: YZ v hypotenuse of the right-angled triangle XZ 4.2 If YZ is the WANTED side and XZ is the GIVEN side : ¥ sin 35,3° 3 ¥o-numerator ; = sin 35,3° v h- denominator 4.3 | Calculate YZ: 4 = sin 35,3° v sin35,3° 5 xz Y ratio YZ ¥ replace values = = sin 35,3° v’simplify 32 v answer YZ = 32sin35.3° = 18,49 cm 44 | Bereken XY: ~ = cos35,3° ¥ cos 35,3° 5 x2 Y ratio XY — = cos35,3° “replace the 32 values XY = 32 cos 35,3° v simplify = 26.12 cm Y answer [16] Vraag5[5] BH Y ° 5 tan 66° = o_ Bn v tan 66 5 a_ BR ratio tan 66° = 22 v replace the 50 values BH = 50.tan 66° ¥ simplify = 112,3m vanswer [5] TOTAL 50

Downloaded from Stanmorephysics.com Level 1 Level 2 Level 3 Level 4 11 15 14 10 22% 30% 28% 20% Angie Of Elevation: itis the angle between a horizontal line from th