The Pythagorean Theorem

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 four angles of the blue four-sided figure to the left all have measure $S + T$, where $\cl"purple"S$, $\cl"purple"T$, and $\cl"purple"{90˚}$ are the measures of the angles of the original triangle. Because the sum of the angles of a triangle is $180˚$, we know that $S + T + 90˚ = 180˚$. What is $S + T$?

The four-sided figure has four right angles, and all of its sides are equal in length (they have length $\cl"blue"c$). Is the four-sided figure a square?

The blue square is made up of four triangles. As you saw in isosceles-qn-1, each of those triangles is congruent to the original triangle, which has an area of $$1/2$$. What is the area of the blue square?

The blue square has side length $\cl"blue"c$, which means its area is $c^2$. You also just found the area of that square. What is the value of $\cl"blue"c$?

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 four-sided 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?

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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:

You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle when you know the lengths of its legs. For example, say you have a right triangle whose legs both have length $1$ and whose hypotenuse has length $c$. What is the value of $c^2$?

Since $c$ is a length, it must be positive. What is the value of $c$?

Does this match what you learned in Questions isosceles-qn-1 and isosceles-qn-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 $c$.

	$a$	$b$	$c^2$	$c$

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 $c^2$.

Is $a^2+b^2=c^2$?

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?