phythagoras solving a missing side of a right angled triangle. miss hudson’s maths
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Phythagoras
Solving a missing side of a
right angled triangle.
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Objective for the next week and a half
- To solve problems involving the Pythagorean Theorem.
For Right Angled Triangles Only!
Hypotenuse - always oppositethe right angle
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Step one - You must always identify which side is the hypotenuse in your
given triangle.
leg
leg
hypotenusea
b
c
This helps us to identify which sides are where in our formula.
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ba
abHypotenuse
c
Pythagorean Theorem2 2 2a b c
In our formula c always represents the longest side known as the hypotenuse and a and b always represent the shorter sides.
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However each right angled triangle can have different letters to identify the sides.
This doesn’t change the formula that we use!
We must still identify the hypotenuse and then the short sides like normal.
B
S
T
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6
8
x
2 2 2a b c 2 2 26 8 x
236 64 x 2100 x
2100 x10 x
x 10 or 10
Solve for x.
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4
7
y
2 2 2a b c Solve for y.
2 2 27 4 y 249 16 y 265 y
265 y
65 y
y 65 or 65 y 8.1
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To the nearest meter, find the length of the diagonal of a rectangle with a width of 4 meters and a length of 10 meters.
4 m.
10 m.
2 2 2a b c 2 2 24 10 x
216 100 x 2116 x
2116 x
x 116
x
x 10.8Miss Hudson’s Maths
20 kms
A car drives 20 kilometres due east and then 45 kilometres due south. To the nearest kilometre, how far is the car from its starting point?
45 kmsx
2 2 2a b c 2 2 220 45 x
2400 2025 x 22425 x
22425 x
x 2425x 49.24
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Written Questions:
A ladder is leaning against a wall. The base of the ladder is 9m away from the base of the wall and the top of the ladder reaches 16m up the wall. What is the length of the ladder?
Hint: Draw a diagram to help your understanding.Give the unknown length of the ladder a pronumeral.
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Written Questions:
On my way to school I walk around a sporting complex. I travel 2.4kms east and then turn south and walk another 3kms. On the way home I decide to take a shortcut through the sporting complex and walk directly home. What distance will I walk when I travel home?
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Written Questions:
In the local Darch park a massive tree has been damaged by a storm. The
Wanneroo shire decides they need to stabilize the tree by attaching wires to
the top of the tree and the ground. If the tree is 17 metres tall and the shire workers want to connect the wires 7.5 meters from the base of the tree how
much wire will they require?
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FINDING THE SHORTER SIDE!
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6
t15
2 2 2a b c
Solve for t.
2 2 2t 6 15 2t 36 225
36 36 2t 1892t 189t 189t 13.7
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Mixed
Problems” -
missing the
hypotenuse or
short side.
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12 cm
5 cmc
c2 = 52 + 122
c2 = 169
c =√169
c = 13 cm
Eg 1: When calculating the length of the hypotenuse, we ADD the squares of the other 2 sides.
Eg 2: When calculating the length of one of the shorter two sides we SUBTRACT the squares.
6 cm
3 cm
x x 2 = 62 - 32
x 2 = 27
x = √27x = 5.2 cm (1dp)
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ApplicationsA ladder is placed up the side of a building. The ladder is 8 m long, and the foot of the ladder is 3 m out from the wall. Draw a diagram to illustrate this, and calculate how far up the building the ladder reaches.
Let x be the height up the building.
x 2 = 82 - 32
x 2 = 55
x = √55 x = 7.42 (2dp)
The ladder reaches 7.42 m up the side of the building.
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A square has a length
of 11mm.Find the length of the
diagonal of the square.
11mm
11mms
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Find the width of a
rectangle with a
length of 20cm and
a diagonal of 24cm.
20cm
w24cm
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A rectangle has a length
of 14cm and a width of
8cm. Find the length of
the diagonals of this
rectangle.14cm
8cm d
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Exte
nsio
n
Questio
nFind the length of a
square which has a
diagonal of 16cm.
Note: Both of the short
sides will be worth the
same value because it is
a square.
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Exte
nsio
n
Questio
nFind the length of a
square which has a
diagonal of 27cm.
Note: Both of the short
sides will be worth the
same value because it is
a square.
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Extension
Question
Find the slant height of a
triangle if the area is
96m2 and has a base of
16m.
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Extension
Question
Find the slant height of a
triangle if the area is
75m2 and has a base of
15m.
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Pyt
hagore
an T
riple
s
3 4 5
6 8 10
9 12 15
12 16 20
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Pythagorean Triples
3 4 5
6 8 10
9 12 15
12 16 20
5 12 13
10 24 26
15 36 39
7 24 25
14 48 50
21 72 75
915
12
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Informal Proof #1
Inscribe a square within the square.
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a b
a
a
a
b
b
b c
c
c
c
2c
Informal Proof #1
2Yellow region c
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a b
a
a
a
b
b
b c
c
c
c
a
b c
2c
Informal Proof #1
2Yellow region c
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b
a
a
a
b
b
c
c
c
a
b c
2c
Informal Proof #1
2Yellow region c
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b
a
a
a
b
b
c
c
c
a
b c
2c
a
bc
Informal Proof #1
2Yellow region c
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b
a
a
b
c
c
a
b c
2c
a
bc
Informal Proof #1
2Yellow region c
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b
a
a
b
c
c
a
b c
2c
a
bc
b
ac
Informal Proof #1
2Yellow region c
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a
b
c
a
b c
2c
a
bc
b
ac
Informal Proof #1
2Yellow region c
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a
b
c
a
b c
2c
a
bc
b
aca
b
c
Informal Proof #1
2Yellow region c
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a
b c
2c
a
bc
b
aca
b
c
Informal Proof #1
2Yellow region c
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a
b c
a
bc
b
aca
b
c
Informal Proof #1
2Yellow region c
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a
b c
a
bc
b
aca
b
c
Informal Proof #1
a
b c
2Yellow region c
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a
bc
b
aca
b
c
Informal Proof #1
a
b c
a
bc
2Yellow region c
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b
aca
b
c
Informal Proof #1
a
b c
a
bc
a
b
c
2Yellow region c
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b
ac
Informal Proof #1
a
b c
a
bc
a
b
c
b
ac
2Yellow region c
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Informal Proof #1
a
b c
a
bc
a
b
c
b
ac
2b
2a
2Yellow region c
2 2Yellow region a b
2 2 2a b c
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a b
a
a
a
b
b
b c
c
c
c
2c
Informal Proof #2
a + bTotal Purple Yellow Area Area Area- =
2(a b) -1
4 ab2
= 2c
2 2a 2ab b 2ab 2c
2 2 2a b c
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