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?