Download 2015-04-xx Similar Right Triangles 4

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SIMILAR RIGHT TRIANGLES #4: SPECIAL RIGHT TRIANGLES
ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE
1. The numbers that our calculator gives us for sine/cosine/tangent are a little bit wonky.
What do you get when you evaluate sin(13o ) ? _________________________ [write out all decimals]
What do you get when you evaluate cos(45o ) ? ________________________ [write out all decimals]
What do you get when you evaluate tan(60o ) ? ________________________ [ write out all decimals]
The decimals that you wrote out… are they exact or approximations? CIRCLE ONE: Exact / Approx.
2. Although all seem mighty random, with wonky decimals, the second and third ones are actually
beautiful numbers that we can easily express, while the first one is just messy and not easily expressible.
Use your calculators to find a decimal approximation for
2
: ______________________
2
Use your calculators to find a decimal approximation for
3 : ______________________
Curious. Very curious. We’re going to understand what is happening.
3. The reason some of the angles have this nice property is that they are related to some nice regular
polygons… An equilateral triangle and a square. Below are diagrams of each. The equilateral triangle has
side length 2t and the square has side length s.1
On the next page are bigger versions. Cut them out, and label all the side lengths and angle measures
that you know.
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If you’re a curious sort, and I hope you are, you probably are wondering why we didn’t give the triangle a side length of t. We could
have, and everything that is to come would have worked out. It just so happens that the algebra works out nicer if we give the
triangle a side length of 2t. That’s all. Nothing special. We aren’t doing any mathematical trickery or lies. If you don’t believe us, do
all the calculations with a side length of t. We’re cool with that.
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4. Now fold the triangle in half – along an angle bisector. We proved previously that this angle bisector will
be the perpendicular bisector of the side opposite of it. So now you have a right triangle!
Label the measure of all the angles that you know.
Label the side lengths that you know.
Without using sine/cosine/tangent, use your brains and some algebra to find the side length(s) that you
don’t know.
Tape your triangle here.
Use your triangle to evaluate exactly:
sin(30o )
cos(30o )
tan(30o )
sin(60o )
cos(60o )
tan(60o )
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5. Now fold the square in half – along a diagonal. Now you have a right triangle!
Label the measure of all the angles that you know.
Label the side lengths that you know.
Without using sine/cosine/tangent, use your brains and some algebra to find the side length(s) that you
don’t know.
Use your triangle to evaluate exactly:
sin(45o )
cos(45o )
tan(45o )
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6. (a) Notice in your 30-60-90 triangle:
The length of the side opposite the smallest angle (30 degrees):
_____ (exact)
The length of the side opposite the middle-sized angle (60) degrees: _____ (exact)
The length of the side opposite the largest angle (90 degrees):
_____ (exact)
If any of your sides involves a radical, next to the exact value write down the decimal approximation of the
length of that side, to the nearest tenth.
(b) Notice in your 45-45-90 triangle:
The length of the side opposite the smaller angles (45 degrees):
_____ (exact)
The length of the side opposite the largest angle (90 degrees):
_____ (exact)
If any of your sides involves a radical, next to the exact value write down the decimal approximation of the
length of that side, to the nearest tenth.
7. (a) If the hypotenuse of a 30-60-90 triangle has a length of 1, find the other two side lengths (exactly).
(b) If the hypotenuse of a 45-45-90 triangle has a length of 1, find the other two side lengths (exactly).
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