Basic Review for the Pythagorean Theorem Lesson

You may be familiar with some of the concepts presented in this and the next page, allowing you to go through them very quickly. However, I hope that you will find some interesting concepts.

Angles and Triangles --

Let's first define an angle. When two lines intersect in a point, called a "vertex", the circular span between the lines is called an angle. The following figure shows the angle n between the lines A and B:

The size of the angle n describes how open or closed the lines are in a circular way. The unit measuring the size of an angle is called a degree, and the symbol ° is used to indicate degrees. How big are the degrees? A full circle of rotation has 360° (as defined by the ancient Greeks). Half a circle has 180°, and half of that, or a quarter of a circle has 90°. The following figure shows a perfect 90° angle between the lines A and B:

Note the small black square between the lines. That special symbol is used to indicate a perfect 90° degree angle, also called a right angle.

Now let's define a triangle. A triangle is a geometric figure consisting of three points or vertices which are connected with straight line segments called sides. When one of the angles of a triangle is a right angle, we call that a right triangle, as shown in this figure:

The letters a, b, and c are normally used to refer to the three sides of a triangle. The letter a stands for altitude and b for base. The c is typically used for the third side, called the hypotenuse. Both sides a and b can be switched; however, the hypotenuse is always the longest side -- the side opposite the right angle, as shown in the figure.

Since you need to understand and state the relationship between the three sides of a right triangle (that is, between a, b, and c in the previous figure), let's look at an example of a relationship. In this example we will be calculating the perimeter or distance around the following rectangle:

To obtain the perimeter p, you need to add the lengths of the four sides or a + b + a + b. This can be expressed in a formula as:

This formula or equation is a relationship, in this case, between a, b, and p. If we know the values of a and b, we can easily calculate the perimeter p. If we know the size of the perimeter and any one of the sides, we can calculate the size of the other side.

Similarly, for the triangle in which we are interested, you need to define a formula which when given the values of a and b, you can calculate the value of c.

What units will we use throughout this lesson? We could use inches or centimeters; however, since we can't really draw things to an exact scale through these computer pages, we will refer to the units simply as "units".

What is an area? It is the number of square units needed to cover a given surface. Suppose that you have a line and that you want to come up with a square that measures exactly the same as the line in both dimensions. In other words, if you have a line that measures 4 units, you want a square that measures 4 units in length and 4 units in width. This is illustrated with the following figure:

You could say that the square on the right corresponds to or was derived from the line on the left. Notice that the line measures 4 units and that the corresponding square measures 4 x 4 = 16 square units (count the small squares on the red square). We can say that when you square a 4 you get 16. The mathematical notation for this is:

The 2 right next to and above the 4 indicates that we must multiply the 4 by itself 2 times such that we have a total of 2 fours, or 4 x 4. If we had a 5 instead of the 2, that would imply multiplying 4 by itself 5 times, or 4 x 4 x 4 x 4 x 4.

We can also do the inverse operation. Suppose that you start with a square that measures 25 square units, and you want to get the line that corresponds to either of its sides (they are both the same). Let's illustrate this with the next figure:

As you can see, we can extract the corresponding line that measures 5 units. In this case we can say that when you UN-square a 25 you get a 5. UN-square is not the correct word, however. The correct word is to take the square root. So the square root of 25 is 5. This is represented in mathematical notation as:

which is the inverse operation of the previous one (you can leave the 2 out and it still indicates a square root).

Since getting the square root of a number is the inverse of squaring a number, both operations cancel each other as in the following two examples:

Let's look at one more example to generalize the concepts. In this case, let's start with a line that is made out of two segments, one measuring a and the other measuring b. The following figure shows the corresponding square derived from the line or the line extracted from the square:

You can see that there are four different areas in the square: two perfect squares of different sizes and two identical rectangles, except for their orientation. A formula to express the size of the square would be:

7² = 49	The square of 7 is 49, and
	The square root of 49 is 7