**What is the Pythagorean theorem?**

He **Pythagoras theorem** List the 3 sides of a right triangle and mention the following:

**The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.**

Look at the Pythagorean theorem in the graph below.

From image:

Legs: a, b (with the sides that form the right angle) Hypotenuse: c (it is the longest side of the right triangle, it is also identified as the opposite side of the right angle)

Then it is fulfilled:

**c² = a² + b²**

**c² = a² + b²**

This **pythagoras formula** is perhaps the most famous of all time and is generally used to find one side of a right triangle, given two of them.

The following video explains what the Pythagorean theorem is with examples.

Next, we will see various **applications of the Pythagorean theorem.**

**Examples of the Pythagorean theorem**

**Example 1:**

**In the following graph, calculate the value of the hypotenuse.**

**Solution**:

Let the hypotenuse be: “x”

Since we are asked for the hypotenuse, we apply the studied theorem:

x² = 5² + 12²

⇒ x² = 25 + 144 = 169

Solving:

**∴ x = 13cm**

**Example 2:**

**In the following figure calculate the value of x**

**Solution:**

In the figure we see that “x” is a leg, we know the other 2 sides, so we apply the Pythagorean theorem:

5² = x² + 3²

Ordering and solving:

x² = 25 – 9 = 16

**⇒ x = 4u**

The triangle with sides 3, 4, and 5 is a Pythagorean right triangle.

**Example 3:**

**In the figure below, find the value of x.**

In this figure the value of x is on all 3 sides, so we quickly apply the Pythagorean theorem:

Solving and factoring we have:

**(x -17)(x – 5) = 0**

From here it can be said that:

** x = 17**

**Example 4:**

**Calculate the height of the building in the following figure.**

**Solution:**

To calculate the height of the building, we must make a “important stroke”, observe:

AP is drawn in such a way that:

ΔAPC = Isosceles Triangle

⇒ AP = 50cm

Therefore, in the right triangle ABP we apply the Pythagorean theorem:

AP² = AB² + BP²

Substituting and solving,

50² = h² + 30²

⇒ h = 40m

**The height of the building is 40m.**

**Example 5:**

**In the following graph, T is the midpoint of AO and P is the point of tangency. Calculate AP.**

**Solution:**

Since P is a point of tangency, we take the opportunity to draw OP and apply the property of the **circumference**:

**OP ⊥ AB**

Be:

AP = x = ??

Let’s see in the graph:

In addition, we see that the right triangle APO is formed, in which ordering the data, we apply the Pythagorean theorem, like this:

AO² = PA² + PO²

Replacing:

4² = x² + 2²

Finally, we give you more examples of the Pythagorean theorem in the following video. Enjoy it!