Pythagoras's Theorem Tuesday, 21 May 2019

Notice 3+4≠5 However… Area = 25 3 5 4 Area = 16 Area = 9 16 + 9 = 25

In general The sum of the areas for the two smaller squares is equal to the area of the larger square . Or in other words the sum of the squares of the two smaller sides is equal to the square of the largest side. Or in algebraic terms The largest side is known as the Hypotenuse b c a

Example Calculate the lettered length in each of the following right angled triangles : (i) c 5 in 12 in 𝑐 2 = 5 2 + 12 2 𝑐 2 =25+144 𝑐 2 =169 𝑐= 169 𝑐=13 in

d 𝑑 2 = 17 2 + 24 2 17 cm 𝑑 2 =865 𝑑= 865 𝑑=29.4 cm (to 1 d.p.) 24 cm (ii) d 17 cm 24 cm 𝑑 2 = 17 2 + 24 2 𝑑 2 =865 𝑑= 865 𝑑=29.4 cm (to 1 d.p.) I point out here that not every right angled triangle produces integer (rational) answers. Hence using the calculator we can reduce the number of lines we write down on paper.

(iv) 𝑒 2 = 15 2 + 8 2 𝑒 2 =289 e 15 cm 𝑒= 289 𝑒=17 cm 8 cm

(ii) f 9 in 12 in 𝑓 2 = 9 2 + 12 2 𝑓 2 =225 𝑓= 225 𝑓=15 in

𝑃𝑄 2 = 15 2 + 21 2 𝑃𝑄 2 =666 𝑃𝑄= 666 𝑃𝑄=25.8 in (to 1 d.p.) Example Find the length of PQ correct to 2 decimal places 21 in 15 in P R Q 𝑃𝑄 2 = 15 2 + 21 2 𝑃𝑄 2 =666 𝑃𝑄= 666 𝑃𝑄=25.8 in (to 1 d.p.)

Find the value of 𝑦 correct to 2 decimal place. Example Find the value of 𝑦 correct to 2 decimal place. 14 cm 9 cm 𝑦 𝑦 2 = 14 2 + 9 2 At this stage I hand out the first homework/classwork task, before moving onto finding a smaller side. 𝑦 2 =277 𝑦= 277 𝑦=16.64 cm (to 2 d.p.)

Pythagoras Theorem Worksheet 1: Finding the Hypotenuse 1. Work out the length of t he hypotenuse for each of the following, giving your answers correct to 1 decimal place. 2 7 6 9 8 4 5 a b c d e f 3 14 13 11 10 21 g h i j (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) Printable worksheet for class or homework

Work out the required lengths for each of the following, giving your answers to 2 decimal places. 2. Find a 3. Find b 11cm 7cm a 5cm b 4. F ind AC 5. Find EF E 12cm D A 9cm 15cm 10cm C B 16cm F 6. Find PR 7. Find LM L R P 4.5m Printable worksheet for class or homework 7cm 7cm M Q 6.3m N

11. Find the length of the diagonal in the rectangle below: 8. Find p 9. Find x 10.5m 9.8m C B 5.9cm 14.3cm p A 17 cm 20cm x 10. Find AC 11. Find the length of the diagonal in the rectangle below: 10 cm 30 cm Printable worksheet for class or homework X Z Y 10 cm 8 cm 12. Find the sloped edge, XY, on the isosceles triangle drawn opposite.

Pythagoras's Theorem Finding a smaller side Start of second lesson. Finding one of the smaller sides of a right angled triangle.

In general If the sum of the squares of the two smaller sides is equal to the square of the largest side. Then the difference between the square of the largest side and the square of one of the smaller sides is equal to the square of the remaining smaller square! In other words The largest side is known as the Hypotenuse b c a

Example Calculate the lettered length in each of the following right angled triangles, giving your answers correct to 2 decimal places. (i) y 14 in 10 in 𝑦 2 = 14 2 − 10 2 𝑦 2 =96 𝑦= 96 𝑦=9.80 in (to 2 d.p.)

(ii) b 36 in 12 in 𝑏 2 = 36 2 − 12 2 𝑏 2 =1152 𝑏= 1152 𝑏=33.94 in (to 2 d.p.)

(vi) e 4.8 in 5.9 in 𝑒 2 = 5.9 2 − 4.8 2 𝑒 2 =11.77 𝑒= 11.77 𝑒=3.43 in (to 2 d.p.)

Example In triangle ABC, angle B = 90⁰ AB = 7in and AC = 11 in. Work out the length of BC, giving your answer correct to 1 decimal place. A 𝐵𝐶 2 = 11 2 − 7 2 11 in 𝐵𝐶 2 =72 7 in 𝐵𝐶= 72 𝐵𝐶=8.5 in (to 1 d.p.) B C

Example Work out the length of NM in the triangle drawn below, giving your answer correct to 1 decimal place 48 in L M N 22 in 𝑀𝑁 2 = 48 2 − 22 2 𝑀𝑁 2 =1820 𝑀𝑁= 1820 𝑀𝑁=42.7 in (to 1 d.p.) I now hand out the second piece of work as homework/classwork before moving onto mixed problems.

Pythagoras Theorem Worksheet 2: Finding a smaller side Printable worksheet for class or homework

Work out the required lengths for each of the following, giving your answers to 2 decimal places. Find m 3. Find p p 15cm m 5cm 11cm 12cm 4. Find AB 5. Find DE A E D 10cm 15cm 25cm C B 8cm F 6. Find PQ 7. Find MN Printable worksheet for class or homework L 14cm R P 8.1m 3.7m 7cm M Q N

L 8. F ind c 9. Find x x c 10. Find JK J K 19cm 14.3cm 5.9cm 13 cm x c 10. Find JK J 10.5m L 5.4m K 11. Find the height of the Isosceles triangle drawn below. S U T h cm 10 cm Printable worksheet for class or homework 16 cm 16 cm

Pythagoras's Theorem Mixed problems Third lesson on Pythagoras looks at students ability to work out whether we square and add or square and subtract.

Example A rope support for a flagpole is attached at a height of 4 m and is fixed to the ground at a distance of 1.4 m from the base. Calculate the length of the rope required, giving your answer correct to 2 decimal place. 4m 1.4m x 𝑥 2 = 1.4 2 + 4 2 𝑥 2 =17.96 𝑥= 17.96 𝑥=4.24 m (to 2 d.p.)

Find the value of 𝑦 correct to 1 decimal place. Example Find the value of 𝑦 correct to 1 decimal place. 14.3 cm 3.7 cm 𝑦 𝑦 2 = 14.3 2 − 3.7 2 𝑦 2 =190.8 𝑦= 190.8 𝑦=13.8 cm (to 1 d.p.)

Calculate the height of the drawn triangle below Example Calculate the height of the drawn triangle below 15 cm 8 cm 15 cm 𝑥 𝑥 4 cm 𝑥 2 = 15 2 − 4 2 𝑥 2 =209 𝑥= 209 𝑥=14.5 cm (to 1 d.p.)

Example A ladder reaches 12m up a vertical wall. If it rests on horizontal ground with the foot of the ladder 1.75m from the foot of the wall, how long is the ladder? 𝑥 2 = 12 2 + 1.75 2 𝑥 𝑥 2 =147.0625 12m 𝑥= 147.0625 𝑥=12.13 m (to 2 d.p.) 1.75m

Example ABCDEF is a prism with a triangular cross-section. 𝐴 𝐵 𝐶=90°, BC = 8m, AB = 6m and CD = 20m. Find (i) AC (ii) BD A F E C B D 6m 20m 8m (i) (ii) 𝐴𝐶 2 = 8 2 + 6 2 𝐵𝐷 2 = 8 2 + 20 2 𝐴𝐶 2 =100 𝐵𝐷 2 =464 𝐴𝐶= 100 𝐵𝐷= 464 𝐴𝐶=10m 𝐵𝐷=21.54 m (2 d.p.)

Pythagoras Theorem Worksheet 3: Mixed problems Answer all the following questions, showing your working. 1. Find EF correct to 1 decimal place. 2. Find p correct to 2 decimal places 22cm 18cm p 4.5cm 11cm d 3. Find d correct to one decimal place. 4. Find BC. 15cm A B C 8cm 7cm 10cm F E D Printable worksheet for class or homework

5. The diagram represents the front end of a garden shed. Find the width of the shed correct to one decimal place. 2.2m 3.1m 2.9m 6. The diagram drawn opposite represents a ladder placed against a wall. Calculate the length of the ladder correct to the nearest centimetre. 2.7m 5.4m Printable worksheet for class or homework 7. Calculate the values of x and y in the diagram below, giving your answers correct to 2 dp. 2.4 m 8 m 6.5 m 4.8 m y x

8. The dotted line on this map represents the journey of a ship travelling from A to D stopping at two ports on route at B and C. Calculate the total length of this ships journey. {answers to one decimal place}. A B C D 1 2 3 4 km 9. Two planes are flying over the village of Colne, one directly above the other when they are picked up by a radar station some 10km away from Colne. The distances of the planes from the radar are given as 13km and 15 km as the diagram shows. Find the distance between the two planes. Colne 13km 15km 10km Printable worksheet for class or homework

Furthermore AB = 4cm, BC = 7cm and BE = 10cm. 10. The diagram below is of a triangular prism with triangle ABC a right angled triangle. Furthermore AB = 4cm, BC = 7cm and BE = 10cm. F A E D B C Find the length of (a) AC (b) AD, giving your answers correct to 2 decimal places. 11. The diagram below represents a cuboid. Find the length of the diagonal PV, giving your answer correct to the nearest whole number. P V W S T R U 17cm 9cm 21cm Printable worksheet for class or homework