Theorem of Pythagoras With Applications
The socalled Theorem of Pythagoras states
that for a right triangle, it is always true that
c^{ 2} = a^{ 2} + b^{ 2}
Recall that in a right triangle, the longest side, opposite
the right angle, is called the hypotenuse. Thus
the Theorem of Pythagoras states that for right triangles
the square of the length of the hypotenuse = the sum
of the squares of the lengths of the other two (shorter) sides
If the lengths of any two sides of a right triangle are known,
then the length of the remaining third side can be computed using
this formula.
(In the examples that follow, we will observe the convention
that for right triangles, the right angle is always labelled C,
and the other two angles are labelled A and B. The labels for the
sides are the lowercase characters of the angles opposite. So,
the side opposite the right angle vertex is always labelled c,
the side opposite vertex A is always labelled a, and the side
opposite vertex B is always labelled b. These symbols are
somewhat loosely used simply as names of these parts of the
triangle, and as well, in formulas, they represent the sizes of
the parts. Thus b is a label for the side opposite vertex B, but
in a formula, the symbol b stands for the length of that side.
Similarly, the symbol B is a label for a vertex, but also, in
formulas represents the size of the angle at the vertex, in some
appropriate unit of measurement such as degrees.)
Example 1:
The two perpendicular sides of a right triangle have lengths
12.3 m and 8.7 m, respectively. Compute the length of the
hypotenuse of this right triangle.
solution:
The situation is shown in the sketch to the right. We are
asked to determine the value of c. Applying Pythagoras’s
Theorem directly gives
c^{ 2} = a^{ 2} + b^{ 2}
= (12.3 m)^{ 2} + (8.7 m)^{ 2}
= 226.98 m^{ 2}
Therefore,
Since the original measurements are rounded to one decimal
place, we should probably do so with our answer. Thus, the length
of the hypotenuse of this right triangle is 15.1 m.
Caution:
People are sometimes tempted to apply Pythagoras’s
Theorem even when the triangle in question is either not a right
triangle or is not known for sure to be a right triangle.
Pythagoras’s Theorem only works for right triangles. If it
is applied in any way to triangles which are not right triangles,
it is sure to give erroneous results.
If we know the lengths of the three sides of a triangle,
Pythagoras’s Theorem can also be used to determine whether
or not that triangle is a right triangle.
Example 2:
Each of the following sets of three numbers are claimed to be
lengths of sides of right triangles. Confirm that this is true in
each case.
(i) 8, 19, 13
(ii) 13, 12, 5
(iii) 7, 42, 9
solution:
In each case, the longest side must be the hypotenuse (since
the hypotenuse is always the longest side of a right triangle).
If the given triangle is really a right triangle, then the square
of the length of the longest side must be exactly equal to the
sum of the squares of the lengths of the other two sides.
So, for case (i), the length 19 must be the length of the
hypotenuse. Here
19^{ 2} = 361.
But
8^{ 2} + 13^{ 2} = 64 + 169 = 233.
Since 233 361, these three numbers do not satisfy
Pythagoras’s Theorem, and so they cannot represent the
lengths of the three sides of a right triangle.
For case (ii), the length 13 must be the length of the
hypotenuse. Then
13^{ 2} = 169
and
5^{ 2} + 12^{ 2} = 25 + 144 = 169
Since these two results are equal, we conclude that these
three lengths do form a right triangle, as sketched in the
figure.
For case (iii), the length 42 would have to be the length of
the hypotenuse. Then, checking, we find that
42^{ 2} = 1764
and
7^{ 2} + 9^{ 2} = 49 + 81 = 130
Since 1764 130, we know that these three lengths
cannot form a right triangle. In fact, since 42 is greater than 7
+ 9 = 16, it turns out that these three lengths cannot even form
a triangle of any shape!
