Loci, circles, trig
Problem 1
There are only a few permutations that we can discuss when talking about loci. The basic objects we have are points, lines, and distance. So we can discuss
a) the locus of points equally distant from a single point
b) the locus of points equally distant from a line
c) the locus of points equally distant from two points
d) the locus of points equally distant from two lines (parallel or non-parallel)
e) the locus of points equally distant from a point and a line
Part (e) is so difficult that you really shouldn't worry about it. But for the others, you should understand what the loci are. Describe all of these from (a) to (d).
Problem 2
What is the locus of points between two parallel planes?
Problem 3
If you have a right triangle with any two sides of a given length, you can find the length of the third side. How?
Below are two examples. Solve the examples and the describe how you can solve any problem like this, regardless of the particular numbers.
Problem 4
If you have a right triangle with any two sides of a given length, and we mark a certain non-right angle, you can find the sine, cosine, and tangent of that angle. How?
Below are examples. Solve the examples and describe how you can solve any problem with any numbers.
Problem 5
Given any side length and angle in a right triangle you can find all sides and all angles. How?
Below are examples. Solve them and describe how a solution goes in general.
Problem 6
The problems above required a calculator to solve. With which angles do you not need a calculator?
Problem 7
If a right triangle has a hypotenuse of 10 and an angle of 40 degrees, what is its area?
Problem 8
Suppose a triangle isn't a right triangle. Still we know two side-lengths and the angle between, like below.
Of course we can't use the usual trig calculations because this isn't a right triangle.
Still it seems that we should be able to figure out the rest of the dimensions of the triangle. This feels like enough information for that to be possible.
Can you do it?
Problem 9
Suppose a regular n-gon has a radius of r. What's its area?
Problem 10
In the figure below, x is the inscribed angle and y is the arc measure from D to B. The arc from D to C is 160. Find x and y.
Problem 11
If in the previous problem the radius were 5, what would the arc length from D to B be?
Problem 12
We've two special cases of angles in circles: The central angle and the inscribed angle. In the first we found that the central angle just is the arc measure of the intercepted arc. In the second case we found that the inscribed angle was half the intercepted arc.
What about other angles? For instance, what if an angle is formed anywhere else inside the circle, like below?
You could draw many pictures like this with the same x value and different y values, so clearly there is no simple relationship between x and y. By that I mean: You need more information than x to learn the value of y. That makes this problem harder than the central or inscribed angle problems.
But what if you knew y and z? Would that be enough information to figure out x?
It turns out the answer is yes and that which is to say that x is the average of y and z.
Suppose x is 100 and z is 40. Find y. Also find the measure of arc EF.
Problem 13
Show that lines g and f are parallel, given that arcs EC and DB are have the same measure.