1. Pythagoras's Theorem
Tuesday, 21 May 2019

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

3. 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

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

5. 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 =
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.

6. (iv)
𝑒 2 =
𝑒 2 =289 e
15 cm
𝑒= 289
𝑒=17 cm
8 cm

7. (ii) f
9 in
12 in
𝑓 2 =
𝑓 2 =225
𝑓= 225
𝑓=15 in

8. 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 =
𝑃𝑄 2 =666
𝑃𝑄= 666
𝑃𝑄=25.8 in (to 1 d.p.)

9. 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 =
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.)

10. 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

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

12. 11. Find the length of the diagonal in the rectangle below:
8. Find p 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.

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

14. 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

15. 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.)

16. (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.)

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

18. 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

19. 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.

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

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

22. 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

23. 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.

24. 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 =
𝑥 2 =17.96 𝑥=
𝑥=4.24 m (to 2 d.p.)

25. 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 = − 3.7 2
𝑦 2 =190.8 𝑦=
𝑦=13.8 cm (to 1 d.p.)

26. 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.)

27. 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 = 𝑥
𝑥 2 =
12m 𝑥=
𝑥=12.13 m (to 2 d.p.)
1.75m

28. 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 =
𝐵𝐷 2 =
𝐴𝐶 2 =100
𝐵𝐷 2 =464
𝐴𝐶= 100
𝐵𝐷= 464
𝐴𝐶=10m
𝐵𝐷=21.54 m (2 d.p.)

29. Pythagoras Theorem Worksheet 3: Mixed problems
Answer all the following questions, showing your working.
1. Find EF correct to 1 decimal place 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

30. 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

31. 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

32. 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