Right Triangles and the Pythagorean Theorem
A Right Triangle is a triangle with a Right Angle, an angle of $\displaystyle 90{}^\circ $, and two acute angles, angles of $ \displaystyle <90{}^\circ $. It makes sense that the other two angles are acute, since the total sum of all angles much equal $\displaystyle 180{}^\circ $.
The Pythagorean Theorem works with Right Triangles and it relatively simple: $ \displaystyle {{a}^{2}}+{{b}^{2}}={{c}^{2}}$, where $ a$ and $ b$ are the measurements of the “legs” of the triangle, and $ c$ is the measurement of the Hypotenuse (the side opposite the right triangle); the converse is also true. Note that the hypotenuse is always the longest side, since it is opposite the largest angle. You may have to determine the third side of a right triangle, given the other two sides.
The most “famous” right triangle is the $ 3$-$ 4$-$ 5$ one, since $ \displaystyle {{3}^{2}}+{{4}^{2}}={{5}^{2}}$. Note that any “multiple” of this triangle also works, like $ \displaystyle {{6}^{2}}+{{8}^{2}}={{10}^{2}}$. Pretty cool!
Here is a list of right triangle sets that you should know off-hand; these are called the Pythagorean Triples. Note again that any of these triples can be multiplied by a number all the way across, and that will also be a Pythagorean triple:
| $(3, 4, 5)$ | $(5, 12, 13)$ | $(8, 15, 17)$ | $(7, 24, 25)$ |
Extension of the Pythagorean Theorem
You can use an extension of the Pythagorean Theorem to determine if a triangle is right, acute, or obtuse:
| Right:
$ \displaystyle {{a}^{2}}+{{b}^{2}}={{c}^{2}}$ |
Acute: $ \displaystyle {{a}^{2}}+{{b}^{2}}>{{c}^{2}}$ |
Obtuse:
$ \displaystyle {{a}^{2}}+{{b}^{2}}<{{c}^{2}}$ |
Special Right Triangles
I always hate to use the words “memorize this”, but, for certain right triangles, there are tricks to use if you memorize some simple things. Don’t worry; if you forget, it’s easy enough to come up with the same tricks.
An isosceles right triangle is one with two equal sides, and thus two equal angles, other than the right angle.
A 30-60-90 right triangle is one with what it says: one 30 degree, one 60 degree, and one 90 degree angle.
The proportions of the sides for each of these are as follows, which can easily be proven:
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