only works in right angled triangles nothing to do with angles
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Only works in right angled trianglesOnly works in right angled triangles Nothing to do with anglesNothing to do with angles
The The hypotenusehypotenuse is is the longest side in a the longest side in a right angled triangle.right angled triangle.
It is always the side It is always the side opposite the right opposite the right angle.angle.
hypotenuse
hypotenusehypotenuse
The area of the square drawn on the The area of the square drawn on the hypotenuse is equal to the sum of hypotenuse is equal to the sum of the area of the squares drawn on the area of the squares drawn on the other two sidesthe other two sides
c2 = a2 + b2
c
b
a
You can visualise You can visualise the theoremthe theorem
b
a c
cc22 == aa2 2 + b+ b22
18
10
?
cc2 2 == 10102 2 + + 181822
Find the missing Find the missing side.side.
100100
324324
100 + 324 = 424100 + 324 = 424
cc22 = 424 = 424
c = √424c = √424
c = 20.6 (1.d.p.)c = 20.6 (1.d.p.)
Give your answers to 1 d.p.
8cm
10cm
11m
7m
24km
5km
cc22 = a = a22 + b + b22
cc22 = 10 = 1022 + 8 + 822
cc22 = 100 + 64 = 100 + 64cc22 = 164 = 164c = √164c = √164c = 12.8cm c = 12.8cm (1.d.p.)(1.d.p.)
cc22 = a = a22 + b + b22
cc22 = 11 = 1122 + 7 + 722
cc22 = 121 + 49 = 121 + 49cc22 = 170 = 170c = √170c = √170c = 13.0m c = 13.0m (1.d.p(1.d.p.)
cc22 = a = a22 + b + b22
cc22 = 24 = 2422 + 5 + 522
cc22 = 576 + 25 = 576 + 25cc22 = 601 = 601c = √601c = √601c = 24.5km c = 24.5km (1.d.p.)(1.d.p.)
181
0 ?
aa2 2 == 18182 2 - 10- 1022
Find the missing Find the missing side.side.
100100
324 – 100 324 – 100 = 224= 224
324324
aa22 = 224 = 224
a = √224a = √224
a = 15.0 (1.d.p.)a = 15.0 (1.d.p.)
18182 2 == aa2 2 + + 101022
Give your answers to 1 d.p.20 cm
12 cm
11m
17m
24km
5km
aa22 = c = c22 - b - b22
aa22 = 20 = 2022 - 12 - 1222
aa22 = 400 - 144 = 400 - 144aa22 = 256 = 256a = √256a = √256a = 16 cm a = 16 cm
aa22 = c = c22 - b - b22
aa22 = 17 = 1722 - 11 - 1122
aa22 = 289 - 121 = 289 - 121aa22 = 168 = 168a = √168a = √168a = 13.0m a = 13.0m (1.d.p(1.d.p.)
aa22 = c = c22 - b - b22
aa22 = 24 = 2422 - 5 - 522
aa22 = 576 - 25 = 576 - 25aa22 = 551 = 551a = √551a = √551a = 23.5km a = 23.5km (1.d.p.)(1.d.p.)
Navigation Navigation problems are often problems are often solved using solved using Pythagoras’ Pythagoras’ Theorem.Theorem.
N
S
EW
A plane leaves an airport and travels 32km west then it turns and travels 41km north. It develops a problem and has to return to the airport. How far is it?
Step 1. Draw a Step 1. Draw a diagramdiagram
32km Airport
?
Step 2. Use Step 2. Use PythagorasPythagoras
41km
cc22 = a = a22 + b + b22
cc22 = 32 = 3222 + 41 + 4122
cc22 = 1024 + 1681 = 1024 + 1681cc22 = 2705 = 2705c = √2705c = √2705c = 52.0km c = 52.0km (1.d.p.)(1.d.p.)
Problems involving Problems involving isosceles triangles isosceles triangles are often solved are often solved using Pythagoras’ using Pythagoras’ Theorem.Theorem.
•Draw Draw perpendicular perpendicular and mark and mark lengthslengths•Use Pythagoras Use Pythagoras theoremtheorem
bb
ac
cc22 = a = a22 + b + b22
A roof on a house that is 6 m wide peaks at a height of 3 m above the top of the walls.
Find the length of the sloping sides of the roof.
3 m
6 m
cc22 = a = a22 + b + b22
cc22 = 3 = 322 + 3 + 322
cc22 = 9 + 9 = 9 + 9cc22 =18 =18c = √18c = √18c = 4.2 m (1.d.p.)c = 4.2 m (1.d.p.)
?? 3 m
3 m
cStep 1. Draw a Step 1. Draw a diagramdiagramStep 2. Use Step 2. Use PythagorasPythagoras
• draw a diagram for the problem that includes a draw a diagram for the problem that includes a right-angled triangleright-angled triangle• label the triangle with the length of its sides label the triangle with the length of its sides from the questionfrom the question• label the unknown side ‘x’ label the unknown side ‘x’ • if it’s the hypotenuse, thenif it’s the hypotenuse, then
“ “SQUARE, SQUARE, SQUARE, SQUARE, ADD,ADD, SQUARE ROOT” SQUARE ROOT”• if it’s one of the shorter sides, then if it’s one of the shorter sides, then
““SQUARE, SQUARE, SQUARE, SQUARE, SUBTRACTSUBTRACT, SQUARE , SQUARE ROOT”ROOT”• round your answer to a suitable degree of round your answer to a suitable degree of accuracyaccuracy
11.6 cm
8.7 cm15.3 cm
C alcu la tions:
D ecis ion: Yes N o
Is this triangle possible?
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