Pythagorean Theorem

The Pythagorean Theorem is used to find a missing side length of a right triangle. It only works for right triangles (one with a right angle).

The formula to use is a² + b² = c². A really good explanation of the formula and its derivation can be found here.

In order to use the theorem, you must know the parts of the triangle. Parts a and b of the theorem are the legs of the triangle. These are the side lengths that make up the right angle. The third side (longest side) is noted as c in the theorem.

Now substitute the values given into the theorem (formula) and solve the equation.

Example: A triangle has a side length of 6 cm and a side length of 8 cm. What is the length of the hypotenuse?

6² + 8² = c²
36 + 64 = c²
100 = c²
√100 = √c²
10 = c

Example: A triangle has a side length of 6 cm and a hypotenuse length of 12 cm. What is the length of the missing side?

a² + 6² = 12²
a² + 36 = 144
a² + 36 – 36 = 144 – 36
a² = 108
√a² = √108
a = √108
a = 6√3

Picture of triangle above from http://en.wikibooks.org/wiki/File:Right_triangle_shows_hyp_legs.PNG

Rate-Time-Distance Word Problems

Or otherwise known as those dreaded train problems. These are always tricky for algebra students. Ask someone you know which problems they remember from Algebra I and almost all of them will have something to say about the train problems.

Train problems can be separated into two categories: same direction travel and opposite direction travel. Each are handled differently, but using a chart makes it easier to set up the equations. You also have to remember that distance equals rate times time or d=rt.

Here is an example. A train leaves the train station at 2:00 p.m. Its average rate of speed is 90 mph. Another train leaves the same station a half hour later. Its average rate of speed is 120 mph. If the second train follows the same route on a parallel track to the first, how many hours will it take the second train to catch the first?

Train	Rate	Time	Distance
1	90	t	90t
2	120	t-0.5	120(t-0.5)

Since the trains are travelling in the same direction, their distances are equal when the 2nd one catches the 1st. Therefore, to solve this equation, we set the distance of Train 1 equal to Train 2.

90t = 120(t-0.5)
90t = 120t - 60
-30t = -60
t = 2

The first train travelled for 2 hours before the 2nd train caught up to it. The problem asks how long it takes the 2nd train to catch the first. So it takes the 2nd train a half hour less than it did the first train which is 1.5 hours.

Check back tomorrow for information on opposite direction travel.

Problems with Starting Homework?

Here's a tip that I use for any job that I continue to put off (procrastinate about). Set a timer for 5, 10 or 15 minutes. You can do anything (even Algebra work) for those periods of time. Sit down and get started. When the timer goes off, stop all work and rest your brain for 10-15 minutes. If there's still more to do, set the timer again and go again. Continue this until you have completed that homework.

Don't forget that there are answers in the back of most books. Checking those answers to see if you are on the right track should not be considered cheating. I encourage my students to check those answers. This will help boost your confidence if you are getting the problems right. It will also encourage you to look back over notes, examples in the textbook, etc. if you are getting the answers incorrect.