$$\definecolor{flatred}{RGB}{192, 57, 43} \definecolor{flatblue}{RGB}{41, 128, 185} \definecolor{flatgreen}{RGB}{39, 174, 96} \definecolor{flatpurple}{RGB}{142, 68, 173} \definecolor{flatlightblue}{RGB}{52, 152, 219} \definecolor{flatlightred}{RGB}{231, 76, 60} \definecolor{flatlightgreen}{RGB}{46, 204, 113} \definecolor{flatteal}{RGB}{22, 160, 133} \definecolor{flatpink}{RGB}{243, 104, 224}$$

## March 6, 2021

### Solving Right Triangles

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You now have everything you need to solve right triangles.

"...wait, what does it mean to "solve a triangle"?" — I hear you ask.

It just means that, given some information about a triangle (angles and sides), you're trying to find any missing pieces.

And with the circular and inverse circular functions, you can do this while starting with a surprisingly small amount of information!

Considering how often triangles show up all over the place just about anywhere you look, this is a useful skill if you want to build or measure things. (Seriously. Try to go a day WITHOUT seeing a triangle somewhere, besides math class.)

So ... without further ado, let's just cut right to the chase!

# Preview Activity 11

You can choose to do EITHER #3 OR #4. You do NOT need to do both.
(...though I won't stop you if you decide to.)

1. Do a quick Google search to find at least three examples of where you think right triangles would be useful in the "real world." See if you can find one you hadn't thought of.
2. A triangle is completely solved when you know all three angles and all three sides. What do you think is the minimum amount of information you'd need in order to be able to completely solve a triangle? Why?
3. Find the angle measurements of a right triangle with each of the following side length measurements. (Draw a right triangle and use inverse circular functions.)
1. $$3, 4, 5$$
2. $$5, 12, 13$$
3. $$9, 12, 15$$
4. Recall that a regular polygon is a polygon whose sides are all the same length and whose angles are all the same measure. (In other words, it's both equilateral and equiangular.)
If a regular $$n$$-sided polygon has side length $$\ell$$, then its area is $$\dfrac14n\ell^2\cdot\cot\left(\dfrac{180^\circ}{n}\right)$$. Show how to derive this formula.
(Hint: Divide it into isosceles triangles, each of which can be split into two right triangles.)
5. Answer AT LEAST one of the following questions:
2. What was an "a-ha" moment you had while doing this activity?
3. What was the muddiest point of this activity for you?
4. What question(s) do you have about anything you've done?