The Pythagorean Theorem: Application

Solving the third side of a right triangle

[br]Suppose we are given a right triangle in a plane, [br][br]and we are asked to find the length of the unknown side with 2 sides given, [br][br]what would you do?

It's pretty easy when we're applying the pythagorean Theorem:

[br][math]3^2+4^2=?^2[/math][br][br][math]9+16=?^2[/math][br][br][math]25=?^2[/math][br][br][math]?=5[/math]

However, what if you get this new problem in your exam? And you don't have a calculator?

Headache !

You will waste quite some time doing them two-digit multiplications! [br][br]And eventually square rooting a large number!

So, is there a better way to solve most of these questions?

Yes ![br]It's all about the similar triangles![br][br]

What are similar triangles?

Similar triangles are the triangles with the same set of ratios between all the sides.[br][br]Namely, the exact same shape.

Example : Similar Triangles

So, how is this helping ?

It turns out that for right triangles to have INTEGER SIDES is a rare thing.[br][br]Since the three sides need to satisfy the Pythagorean Theorem: [math]a^2+b^2=c^2[/math][br][br]So it's possible to classify the classic triples by sorting together the similar ones!

Example: The classic (3,4,5) triple

Now that we look at this question, what should we do?

We can identify the class of this triangle first!

We want to zoom in or zoom out the ratio of the three sides,[br][br]in order to get the same ratio, [br][br]but with the smallest possible integer set !

Now we look at this question, we can do this now:

After realizing the unknown side, remember to zoom back to the original ratio!

In the smallest ratio, it's easy to recognize the triple and get the unknown side.[br][br]But remember! You got to zoom back to the original ratio!

After recognizing the (3,4,5) triple, we got to enlarge it back to its original ratio!

Example 2:

Now that's just (3,4,5), what about the other triples?

So it turns out that they have a name: Pythagorean triples.[br][br]And, they are pretty rare![br][br]It's easy to remember the smallest ones, and it'd be super useful in exams!

The most common triples in exams:

Should we memorize all the Pythagorean triples?

There are infinitely many Pythagorean triples, so it's impossible to memorize all![br][br]But, it's very easy to memorize the smallest few, and this will be extremely helpful already![br][br]In exams, often times we wouldn't encounter right triangles with length of decimals,[br][br]and if we see an unknown triple, [br][br]it's time to go back and use the formula [math]a^2+b^2=c^2[/math], as all people without a calculator will have to do so.[br][br]

Conclusion: Try to identify first, then use the formula

Next time we have a similar question,[br][br]don't apply the formula right away, [br][br]try to identify the triangle first,[br][br]then zoom back to the solution !