(This lesson also appears in the Algebra I curriculum.
If you studied it there, then just quickly review, and move on to the next section!)
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
[beautiful math coming... please be patient]$\,\,T\,\,$
be a triangle with sides of lengths
[beautiful math coming... please be patient]$a$,
[beautiful math coming... please be patient]$b$, and
[beautiful math coming... please be patient]$c$,
where
[beautiful math coming... please be patient]$\,c\,$ is the longest side (if there is a longest side). Then,
[beautiful math coming... please be patient]$T\,$ is a right triangle
if and only if
[beautiful math coming... please be patient]$a^2 + b^2 = c^2\,$.


Have fun with
many proofs of the Pythagorean Theorem!
This applet (you'll need Java) is one of my favorites. Let it load, then keep pressing ‘Next’.
EXAMPLES:
Question:
Suppose that two angles in a triangle are 60° and 30°.
Is it a right triangle? Answer YES, NO, or MAYBE.
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:
[beautiful math coming... please be patient]$3^2 + x^2 = 5^2$
[beautiful math coming... please be patient]$9 + x^2 = 25$
[beautiful math coming... please be patient]$x^2 = 16$
[beautiful math coming... please be patient]$x = 4$
Note:
[beautiful math coming... please be patient]$x\,$ cannot equal
[beautiful math coming... please be patient]$\,\text{}4\,$, because lengths are always positive.
The
345 TRIANGLE is a wellknown 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
345 triangle by any positive real number
k ,
then you will still have a right triangle.
For example, these are all right triangles:
6810 
( k = 2 ) 
91215 
( k = 3 ) 
1.522.5 
( k = 0.5 ) 
3π4π5π 
( k = π ) 
and so on!
Question:
Suppose a triangle has sides of lengths
[beautiful math coming... please be patient]$1\,$,
[beautiful math coming... please be patient]$\sqrt{3}$, and
[beautiful math coming... please be patient]$2\,$. Is it a right triangle?
Solution:
YES.
Since
[beautiful math coming... please be patient]$2 > \sqrt{3}\,$,
the longest side has length 2.
And,
[beautiful math coming... please be patient]$1^2 + {(\sqrt{3})}^2 = 1 + 3 = 4 = 2^2\,$.