98 pythagoras’ theorem pythagoras’ theorem 98 · angle (the longest side of the triangle) is...
TRANSCRIPT
SKILL Use Pythagoras’ theorem to fi nd one side of a right-angled triangle, given the lengths of the other two sides
EXAM FACTS
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KEY FACTS
In a right-angled triangle the side opposite the right angle (the longest side of the triangle) is called the hypotenuse of the triangle.
In the diagram the length of the hypotenuse is c.
Pythagoras’ theorem states that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
That is c2 = a2 + b2
In triangle DEF, Pythagoras’ theorem gives DE2 = EF2 + DF2
DE2 means that the length of the side DE is squared.
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b
a
c
F E
D
Getting it right
In triangle DEFFE = 8.7 cm,DF = 6.4 cm,Angle DFE = 90°.Calculate the length of DE.Give your answer correct to 1 decimal place.
DE2 = EF2 + DF2
DE 2 = 8.72 + 6.42
DE 2 = 75.69 + 40.96
F E
D
6.4 cm
8.7 cm
Diagram NOT accurately drawn
EXAM TIP“Diagram NOT accurately drawn” means that taking measurements from the diagram will not give the correct answer.
Identify the hypotenuse of the triangle (the side
opposite the right-angle). Then write down Pythagoras’
theorem for the triangle.
Substitute the given lengths. You would get 1 mark
for this.
33 Pythagoras’ theorem
96 Pythagoras’ theoremAIMH_C33.indd 96 13/6/07 09:00:29
Now try theseIn Questions 1–4, work out the lengths of the sides marked with letters. The diagrams are not accurately drawn.Give each answer correct to 1 decimal place.
1 2 3 4
DE2 = 116.65
DE = 116.65 = 10.80046…
DE = 10.8 cm
7 cm
12 cm
a10.3 cm
b 4.6 cm
4 cm
c
9 cm
5.8 cm
13.6 cmd
WARNINGA common error is to fail to fi nd the square root and give the answer as 116.65
!
Remember to round your answer to 1 decimal place
and write the units.
PQR is a right-angled triangle.Angle PQR = 90°. QR = 15 cm. PR = 19 cm.Work out the length of PQ.Give your answer correct to 1 decimal place.
(1388 November 2005)
PR2 = PQ2 + QR2 192 = PQ2 + 152
361 = PQ2 + 225 361 − 225 = PQ2
PQ2 = 136 PQ = 136 = 11.6619...
PQ = 11.7 cm
P Q
R
19 cm 15 cm
Diagram NOT accurately drawn
WARNINGThe side to be found is not opposite the right-angle so it is NOT the hypotenuse. A common error is to write incorrectly PQ2 = 152 + 192 This gives PQ = 24.2 which is not sensible, as PQ must be shorter than the hypotenuse, PR.
!
Pythagoras’ theorem 97 AIMH_C33.indd 97 13/6/07 09:00:32
5 PQR is a right-angled triangle. PR = 6 cm. QR = 4 cm Work out the length of PQ. Give your answer correct to 1 decimal place.
(1387 June 2006)
6 In triangle PQR QR = 9.3 cm. PQ = 5.7 cm. Angle PQR = 90°. Calculate the length of PR. Give your answer correct to 1 decimal place.
(1388 November 2005)
7 Work out the value of x.
(4400 May 2006) 8 ABC is a triangle. AB = AC = 13 cm. BC = 10 cm. M is the midpoint of BC. Angle AMC = 90°.
Work out the length of AM. (4400 November 2006)
9 The diagram shows three cities. Norwich is 168 km due East of Leicester. York is 157 km due North of Leicester. Calculate the distance between Norwich and York. Give your answer correct to the nearest kilometre.
(1387 November 2006)10 The diagram shows the positions of three
telephone masts A, B and C.
Mast C is 5 kilometres due East of Mast B. Mast A is due North of Mast B and 8 kilometres from Mast C.
Calculate the distance of A from B. Give your answer in kilometres, correct to
2 decimal places.
(1385 June 1999)
5.7 cm
9.3 cmQ R
P
Diagram NOT accurately drawn
7.5 cm x cm
7.2 cm
Diagram NOT accurately drawn
13 cm
10 cm
13 cm
A
B M C
Diagram NOT accurately drawn
York
Leicester Norwich
157 km
168 km
Diagram NOT accurately drawn
N
N
A
B C5 km
8 km
Diagram NOT accurately drawn
4 cm
6 cm
RQ
P
Diagram NOT accurately drawn
Pythagoras’ theorem 98 Pythagoras’ theorem 98 98 Pythagoras’ theoremAIMH_C33.indd 98 13/6/07 09:00:33