We know from an earlier post that a number shows where you are on a number line. For example, we know the number 7 tells we are 7 units to the right of 0. The number -9 tells us we’re nine units to the left of zero.

What happens if we need to locate a point against two number lines, though. Let’s call our normal number line the “x-axis”. Then, let’s create another number line that intersects the x-axis at zero on both lines, and call this new vertical number line the “y-axis”.

Now we need to describe a point in relation to both axes. Let’s start with that point 7 units away from 0 on the x-axis. That point could be on the x-axis, or it could be somewhere above or below the line. That’s where the y-axis comes in. It tells us how many units above or below the x-axis the point is. For example, our point could be located 3 units below the x-axis, making its location on the y-axis -3.

So our point is at 7 on the x-axis and -3 on the y-axis. That’s a lot of words to describe where this point is located, so we can use a shorter notation called the ordered pair.

The basic pattern for an ordered pair is (x, y).

Since we know our values for both axes, we can plug the 7 in for x and the -3 in for y to come up with the ordered pair (7, -3)

The important thing to remember with an ordered pair is that the first number always describes where you are in relation to the x-axis, and the second describes where you are in relation to the y-axis.

On the bulletin board that plays home to things the students give me, there is one large drawing that will never come down. His name is George, and the band on his hat reads “SOHCAHTOA”. George was a student’s means of remembering basic trigonometric functions.

While none of my other students have met George, most of them have a page to remind them what SOHCHTOA stands for. This mnemonic will help you keep the trig functions straight. In order, SOHCAHTOA means:

sine = $latex \frac{opposite}{hypotenuse}$ cosine = $latex \frac{adjacent}{hypotenuse}$ tangent = $latex \frac{opposite}{adjacent}$

That’s great, but what does it mean?

Opposite and adjacent describe the legs of the triangle in relation to the angle you’re working with. The leg not touching the angle is the “opposite” leg. The leg touching the angle is the “adjacent” leg. The hypotenuse will always be the longest side of the triangle. To solve, find the formula for the function you’re working with and plug in the appropriate numbers.

If you have the measures of two sides, but need the third, you can use the Pythagorean Theorem to find the missing side.

If you’ve been doing math for any period of time, you’ve probably run into a formula that looks like this:

a² + b² = c²

This very useful bit of math is called the Pythagorean Theorem, named after Greek mathematician Pythagoras. Put into words, the above equation tells us that the sum of the square of the two legs of a right triangle equals the square of the hypotenuse (the longest side of a right triangle).

This formula has many potential uses. If you know the length of both legs, or one leg and the hypotenuse, of a right triangle, then you can solve for the missing side using the Pythagorean Theorem.

For example, we are given that a = 3 and b = 4. Let’s solve for c.

3² + 4² = c²

9 + 16 = c²

25 = c²

$latex \sqrt{25}$ = c

5 = c

You can also use the Pythagorean Theorem to prove whether or not a triangle is a right triangle. For this example, let’s have a = 5, b = 10, and c= 13.

5² + 10² = 13²

25 + 100 = 169

125 = 169

Wait a second! 125 does not equal 169. Therefore, a triangle with sides 5, 10, and 13 is not a right triangle.

Let’s try it again with a triangle with the sides 7, 24, and 25.

7² +24² = 25²

49 + 576 = 625

625 = 625

This triangle is a right triangle!

In a bind where a right triangle is involved? Try out the Pythagorean Theorem and see if it helps!