A special example
Let’s start by looking at a specific right triangle. In the
triangle drawn to the left, both of the sides which are next to the
right angle have length $\cl"red"1$. The other side has length
$\cl"blue"c$, and we would like to figure out the value of $\cl"blue"c$.
What is the area of this triangle?
(Notice that it is half of a square with side length 1, or use the formula
$A={1/2}bh$ for the area of a triangle.)
 
Click . Now you can see a figure made up of
four triangles like the first one. Each of these triangles has a Side of length
$\cl"red"1$, a right Angle, and then another Side of length $\cl"red"1$.
Which congruence rule (SAS, ASA, SSS)
relates each of these triangles to the original triangle?
 
Are each of these new triangles congruent to
the original triangle?
 
What is the length of the blue side of
each new triangle? (Your answer may be a variable.)
 
The Pythagorean Theorem
In a right triangle where the two sides next to the right angle (the
legs) each have length 1, you now know that the third side (the
hypotenuse) has length $√2$. Next we’ll look at some more right
triangles, and try to come up with a way to find the length of the hypotenuse if we know the
lengths of the two legs.
In the red right triangle to the left, the legs have
lengths $a$ and $b$, and the hypotenuse has length $c$. Let’s see if we can find a relationship
between $a$, $b$, and $c$.
Click . In the figure to the left, four colored triangles
like the red triangle have been drawn inside a
square with side length $a+b$. Each of these triangles has a Side of length $a$, a
right Angle (which is one of the corners of the square), and then another Side of
length $b$.
Which congruence rule (SAS, ASA, SSS) relates each of these triangles
to the original triangle?
 
Are each of these new triangles congruent to the original triangle?
 
What is the length of the hypotenuse of
each new triangle? (Your answer may be a variable.)
 
We would like to understand the white foursided figure in the middle of
the picture to the left. As you have seen, each of its sides has length $c$. Also, because the
colored triangles have angles which measure $S$, $T$, and $90˚$, we know that
$S+T+90˚=180˚$.
Each angle of the white figure, when added to an angle of
measure $S$ and an angle of measure $T$, forms a straight line (180˚ angle). Since
$S+T+90˚=180˚$, what is the measure of each angle of the white figure?
 ˚ 
The white figure has four right angles, and all of its sides
are equal in length (they each have length $c$). Is the white figure a square?
 
What is the area of the white square?
 
Two squares of side length $a+b$ are drawn to the left, split up in two different
ways. The top square is split up in the same way as in the last two questions; the bottom square
is split up into a square of side length $a$, a square of side length $b$, and four colored
triangles.
Each of the colored triangles in the bottom square has a Side
of length $a$, a right Angle, and then another Side of length $b$. Which
congruence rule relates each of these triangles to the colored triangles in the top square?
 
In the bottom figure, what is the area of the top left white
square (with side length $a$)?
 
In the bottom figure, what is the area of the bottom right white
square (with side length $b$)?
 
The four colored triangles in the bottom square are congruent to the four colored triangles
in the top square, so their areas are equal. Also, the two squares both have side length $a+b$,
so their total areas are equal. So the remaining white areas must also be equal. Click
to mark these white areas. This
demonstrates:
The Pythagorean Theorem
If a right triangle has legs with lengths $a$ and $b$, and a hypotenuse
with length $c$, then
$$
a^2+b^2=c^2
$$
Each row in the table below gives the lengths $a$ and $b$ of the legs of a right
triangle (pictured to the left). Find the length $c$ of the triangle’s hypotenuse using the
formula $a^2+b^2=c^2$ from the Pythagorean Theorem. If $c^2$ is a perfect
square (that is, the square of a whole number), give the exact value of $c$.
The converse to the Pythagorean Theorem
In the last section, you learned that if $a$, $b$, and $c$ are the side lengths of
a right triangle with $c$ the length of the hypotenuse, then $a^2+b^2=c^2$. Now let’s say $a$,
$b$, and $c$ are the side lengths of any triangle. If $a^2+b^2=c^2$, does that triangle
have to be a right triangle?
The red triangle drawn to the left has sides
with length $a=6$, $b=8$, and $c=10$. Compute the value of $a^2+b^2$.
Now compute the value of $c^2$.
 
Click to see a blue right
triangle drawn below the red triangle. The
blue right triangle has legs of length $6$ and $8$, equal in length to the two shorter
sides of the red triangle.
Using the Pythagorean Theorem, find the length of the hypotenuse of
the blue triangle ($\cl"blue"c$).
 
Are the lengths of all the Sides of the
blue right triangle equal to the lengths of all the Sides of
the red triangle?
 
Which congruence rule (SAS, ASA, SSS) relates the
red triangle to the blue right triangle?
 
Is the red triangle a right triangle?
(Hint: Remember that congruent triangles have the same angle
measures.)
 
If you start with any triangle with side lengths $a$, $b$, and $c$, and $a^2+b^2=c^2$, you
can build a right triangle which is congruent to it in this way. This is called the
converse to the Pythagorean Theorem:
If a triangle has side lengths $a$, $b$, and $c$, and $a^2+b^2=c^2$, then
it is a right triangle, and the side with length $c$ is its hypotenuse.
Each row of the table below gives three numbers $a$, $b$, and $c$ which are the
lengths of the sides of a triangle. Using scratch paper, determine whether that triangle is a
right triangle, with $c$ the length of its hypotenuse.
 $a$  $b$  $c$  $a^2+b^2$  $c^2$  Is it a right triangle?
