The Pythagorean Theorem

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THE PYTHAGOREAN THEOREM

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A 90° angle is called a right angle.
A right triangle is a triangle with a 90° angle.

In a right triangle, the side opposite the 90° angle is called the hypotenuse
and the remaining two sides are called the legs.

The angles in any triangle add up to 180°.

In any triangle, the longest side is opposite the largest angle,
and the shortest side is opposite the smallest angle.
Thus, in a right triangle, the hypotenuse is always the longest side.

The Pythagorean Theorem gives a beautiful relationship between the lengths of the sides in a right triangle:
the sum of the squares of the shorter sides is equal to the square of the hypotenuse.
Furthermore, if a triangle has this kind of relationship between the lengths of its sides, then it must be a right triangle!

THE PYTHAGOREAN THEOREM

Let T be a triangle with sides of lengths a , b , and c ,
where c is the longest side (if there is a longest side). Then,

T is a right triangle if and only if a² + b² = c² .

Click here for a geometric proof of the Pythagorean Theorem.

EXAMPLES:

Question: Suppose that two angles in a triangle are 60° and 30°.
Is it a right triangle?
Solution: Yes.
The third angle must be 180° - 60° - 30° = 90°.

Question: Suppose that a triangle has a 100° angle.
Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: No.
The remaining two angles must sum to 80°, so neither remaining angle is a 90° angle.

Question: Suppose that a triangle has a 70° angle.
Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: Maybe.
The remaining two angles must sum to 110°, so one of the remaining angles could be a 90° angle.

Question: Suppose the legs of a right triangle have lengths 3 and x ,
and the hypotenuse has length 5 . Find x .
Solution:
3² + x² = 5²
9 + x² = 25
x² = 16
x = 4
Note: x cannot equal -4, because lengths are always positive.

The 3-4-5 TRIANGLE is a well-known right triangle.
Multiplying all the sides of a triangle by the same positive number does not change the angles.
Thus, if you multiply the sides of a 3-4-5 triangle by any positive real number k ,
then you will still have a right triangle.
For example, these are all right triangles:
6-8-10    ( k = 2 )
9-12-15    ( k = 3 )
1.5 - 2 - 2.5    ( k = 0.5 )
3π-4π-5π    ( k = π )
and so on!

Question: Suppose a triangle has sides of lengths 1, 3 , and 2. Is it a right triangle?
Solution: YES.
Since 2 > 3 , the longest side has length 2.
And, 1 2 + ( 3 ) 2 = 1 + 3 = 4 = 2 2 .

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.