Download tr1.1: pythagoras` theorem

TR1.1: PYTHAGORAS’ THEOREM
Right-angled triangles
A right-angled triangle has an angle of 90o. In a right-angled triangle the side opposite the right angle is called
the hypotenuse. It is also the longest side.
hypotenuse
Right angle
(90o)
Pythagoras’ Theorem
Pythagoras’ theorem states:
In a right angled triangle the square of the length of the hypotenuse (h), is equal to the sum of the squares of the
other two sides.
h
b
2
h=
a 2 + b2
a
Pythagoras’ Theorem can be used to find a side length of a right angled triangle given the other two side
lengths.
Example
Finding the hypotenuse
Find the length of the hypotenuse (h) in the following triangle.
Use Pythagoras theorem h= a + b
with a = 6cm. and b = 8 cm.
2
h
6 cm
2
2
h 2 = 62 + 82
∴ h 2 = 36 + 64
∴ h 2 = 100
∴ h = 10
Note:
Measurements must be in the
same units and the unknown
length will be in these same
units - so h will be 10 cm.
8 cm
TR1.1 - Trigonometry: Pythagoras’ Theorem
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June 2012
Finding a shorter side
Find the value of x in the triangle below.
In this triangle, one of the shorter sides is to be
calculated.
2
h=
a 2 + b2
x
2.7
h = 4.2, a = 2.7 and b = x
2
=
4.2
2.7 2 + x 2
∴ 4.22 − 2.7 2 =
x2
∴17.64 − 7.29 =
x2
4.2
∴10.35 =
x2
∴x =
3.22
or
rearrange Pythagoras’ theorem from
h= a + b 2
2
2
to
2
a=
h2 − b2
where a is the unknown side.
Using the above triangle x2 = 4.22 – 2.72 giving x = 3.22 as before.
Pythagorean triples
In some right-angled triangles all three sides have integer values. These three values form a Pythagorean triple.
Some examples are triangles with sides: (3,4,5), (5,12,13), (7,24 25) and (8,15,17) Check!
Multiples of these, such as (6,8,10) and (9,12,15) are also Pythagorean triples.
Exercises
Find the missing sides in the following.
(1)
9mm
(2)
a
h
20.2
6.5
12mm
TR1.1 - Trigonometry: Pythagoras’ Theorem
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June 2012
(3)
(4)
7cm
4.8 cm
a
c
(5)
14 cm
6.2 cm
(6)
25
37
b
34
a
12
Answers
(1) 15mm.
(2) 19.13
(3) 7.84 cm.
(4) 12.12 cm.
TR1.1 - Trigonometry: Pythagoras’ Theorem
(5) 42.2
(6) 35
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June 2012