The Pythagorean Theorem

The Pythagorean Theorem is perhaps the most famous mathematical theorem ever. You can read about Pythagoras of Samo elsewhere. Now let's get to the triangles.

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area of the square over the side of a triangle

The Pythagorean Theorem tells us a useful relationship between the area of squares that we construct using the sides of a right triangle. Imagine we have this right triangle:

The square grid in the background gives us the dimensions of the triangle. The length of the AC branch is equal to 4 and the length of the BC branch is equal to 3. But what is the length of the side AB, the length of the hypotenuse of the triangle? We can't find out the length at a glance. But we can try to calculate it.

First we construct a square over the hypotenuse BC. By this we mean that we construct a square that has the length of all its edges equal to the length of the side BC, i.e. three. We would draw it in the figure as follows:

We have created a square BCDE, which has the length of all edges equal to 3. What is the area of such a square? We calculate the area of the square by multiplying the lengths of the two edges of the square, so that the area of the square BCDE is equal to 3 · 3 = 9. Draw a square in the figure above the second tangent, above the tangent AC. This square will have an edge length of 4.

This square will have the content 4 · 4 = 16. Now draw the last square, above the overhang AB:

We don't know what the area of this new square will be yet, because we don't know the edge length of AB. But now we can make nice use of the Pythagorean theorem. Namely, it says that the area of this last square ABHI is equal to the sum of the area of the previous two squares. So the first two squares had area of 9 and 16, and the last square - according to the Pythagorean theorem - has the content 9 + 16 = 25. Let's draw in the figure:

And now the last question remains to be answered - if the square ABHI has the content 25, what is the length of the sides of this square? It must be the square root of 25, so the side length of AB is equal to $\sqrt{25}=5$. If we check back and calculate the area of a square with edge length 5, we get the area of 5 · 5 = 25. And now we can get down to numbers and definitions.

The definition of

The Pythagorean theorem goes something like this, "The area of the square over the hypotenuse of a right triangle is equal to the sum of the area of the squares over its branches". Mathematically, this theorem is usually written like this

$$c^2 = a^2 + b^2,$$

where a and b are the lengths of the branches in the triangle and c is the length of the hypotenuse.

So what does the Pythagorean theorem say and how does this notation relate to the previous figure? The area of a square over a side in a triangle means that we take a square that has a side length equal to the length of that given side and calculate its content. We calculate the area of the square by multiplying one side length by the other side length, so if a side has length c, the area of S will be equal to: S = c · c, which we can write as S = c² using powers of 1.

Thus the notation c² = a² + b² can be read just as "the area of a square with edge length c is equal to the sum of the area of squares with edge lengths a and b".

Once again, I'll point out that the theorem only holds in a right triangle, not a general triangle. The Pythagorean theorem is classically used when you know the size of two sides and need to calculate the length of the remaining side. Thus, if we know the length of the two branches a and b and want to get the length of the hypotenuse c, then we compute the contents over the branches, i.e. we compute a² + b². This gives the area of the square above the hypotenuse c, i.e. we get c². To get the side length c, we just subtract the calculated content. This gives us the formula:

$$c=\sqrt{a^2+b^2},$$

where c is the length of the hypotenuse and a, b are the lengths of the offsets. If, on the other hand, we knew the length of the hypotenuse and one of the branches and wanted to calculate the length of the remaining branch, we would calculate it the same way, but we would first isolate one of the branches in the given equation. So if we know c and b and we want to calculate a, then in Eq.

$$c^2 = a^2 + b^2,$$

we isolate a² by subtracting b²:

$$c^2 - b^2 = a^2,$$

swapping the left and right sides:

$$ a^2=c^2-b^2 $$

and finally subtract:

$$ a=\sqrt{c^2-b^2} $$

Example one

Consider the triangle ABC, for which we know the lengths of two sides: a = 3 and b = 4. This is the same triangle we had at the beginning. The question is, what is the length of the remaining side, the side c? Let's try to calculate it using the above formula. The triangle is shown in the following figure:

We can see that we know the lengths of the two branches and we are missing the length of the hypotenuse. Therefore we use the first, unmodified formula:

$$c=\sqrt{a^2+b^2}$$

We add the lengths of the sides after the variables a and b:

$$c=\sqrt{3^2+4^2}$$

In the formula, we have the squared term - as a reminder, the following relation holds:

$$a^2=a\cdot a\quad\rightarrow\quad4^2=4\cdot4=16$$

So, after calculating the square root, we get:

$$\begin{eqnarray} c&=&\sqrt{3^2+4^2}\\ c&=&\sqrt{9+16}\\ c&=&\sqrt{25}\\ c&=&5 \end{eqnarray}$$

As a result, the side length of c is equal to five.

The second example

In this example, we will try to calculate the length of one of the branches if we know one branch and the length of the hypotenuse. So we have a triangle ABC with side lengths |AB| = 10 and |BC| = 6. The triangle is shown in the following figure:

We can see from the figure that the hypotenuse, the longest side, is not the side |AB|. We need to calculate the length of the side |AC|. We plug in the second formula to calculate the length of the branch:

$$a=\sqrt{c^2-b^2}$$

To use the Pythagorean theorem correctly, we now need to understand what each variable stands for. In this formula, the variable a is the length of the side we want to calculate, in this case the side |AC|. The variable c is the length of the hypotenuse, the longest side, in this case the side |AB|. And the variable b is the length of the branch whose length we know, in this case |BC|. Therefore, we plug into the formula as follows:

$$|AC|=\sqrt{|AB|^2-|BC|^2}$$

We add the specific side lengths that we know:

$$|AC|=\sqrt{10^2-6^2}$$

Calculate the powers and subtract:

$$|AC|=\sqrt{100-36}=\sqrt{64}$$

And finally, subtract:

$$|AC|=8$$

The length of the side |AC| is equal to eight.

The word problem

Imagine you are walking to your friend's house on a straight path. The path is 250 metres long. After this quarter of a kilometre you turn left and walk another 100 metres and you are at your good friend's house. The question is, how much shorter will the path be if you take the straight path across the field?

As a first step, calculate the length of the path if you walk along the road. You go first 250 metres straight ahead and then 100 metres to the left. That's a total of 250 + 100 = 350 metres. Now it's time to calculate the length of the path across the field. For this, we'll need a picture.

One piece in the picture represents 50 metres. What would a straight path across the field look like? It would be a line from the point |A| to the point |C|. Let's draw it in red in the figure.

Now we can see that we just need to apply the Pythagorean theorem appropriately to calculate the length of the path across the field. We are looking for the length of the hypotenuse, and we know the length of the two branches, so we use the first formula and add to it as follows:

$$|AC|=\sqrt{|AB|^2+|CB|^2}$$

We add the lengths:

$$|AC|=\sqrt{250^2+100^2}$$

Amplify:

$$|AC|=\sqrt{62500+10000}=\sqrt{72500}$$

Subtract and round:

$$|AC|=269$$

The length of the path across the field is 269 metres. That's not the end of the example, the question was how much shorter is the path across the field. So subtract the lengths from each other, mark the difference r:

$$r=350-269=81$$

The answer is that the path is shorter by approximately 81 metres.

The length of the side of the square

What is the side length of a square that has a diagonal length of ten? How does the Pythagorean theorem help us to do this? We need to find a right triangle in the square to use to calculate the length of the side of the square. Just as a reminder, a square has four equal side lengths. Let's draw the current situation:

We can see that there are suddenly two right-angled triangles in the square, which we could use to calculate the side length. For example, consider the triangle BCD. The side DB is the hypotenuse, the other two are the hypotenuse. We don't know the lengths of the hangers, but we do know the length of the hypotenuse. These branches have the same length, i.e. |BC| = |CD|. We use the basic Pythagorean theorem:

$$|BD|^2=|BC|^2+|CD|^2$$

But since the offshoots are equal, we just need to calculate the square of one offshoot and multiply that by two - we don't need to calculate the square of the two offshoots because they are equal and we would get the same result. So we can write:

$$|BD|^2=2\left(|BC|^2\right)$$

We know the length of the side BD, that's the hypotenuse. The length is equal to ten. We plug it into the equation:

$$10^2=2\left(|BC|^2\right)$$

Let's factor ten:

$$100=2\left(|BC|^2\right)$$

Divide by two:

$$\frac{100}{2}=|BC|^2$$

We truncate the fraction:

$$50=|BC|^2$$

Now we're almost there. We know that the square's side length squared is equal to 50. So to find the side length, we still have to square the equation, which we can afford to do since we are in the positive numbers:

$$\sqrt{50}=\sqrt{|BC|^2}$$

We leave the square root of fifty in this form, but we can cancel the square root and the power on the right side of the equation because they cancel each other out.

$$\sqrt{50}=|BC|$$

If you want, you can square root the 50th if you want. In round numbers, you'd come up with seven:

$$7=|BC|$$

So the length of the side of the square is approximately seven, exactly the square root of fifty.