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Say we have a theatre which has a screen that is two feet off the ground and four feet tall. We wish to see the picture under the largest vertical viewing angle while laying down. How far away should our head be position horizontally from the screen? Our situation looks something like the following:
The red lines represent the angle that the viewer is viewing the screen at.
Now, we would like to be able to talk about this with more precision. Let's use a grid where each unit is one foot. We shall label our points A, B, and C.
Notice that the bottom of the screen is (0, 2) or A, the top of the screen is (0,6) or C, and the viewer's head is B which is not predetermined. We wish to maximize angle ABC.
Now we recall the inscribed angle theorem. This theorem states that If A, B, and C are points on a circle and O is the origin, then 1/2angle AOC = angle ABC. This should give us insight into where the viewer's head should be, since it would allow us to relate our desired angle to an angle from the origin.
First, we determine the circle which has these points. The origin can be found by taking the intersection of two perpindicular bisectors. This intersection is our origin because perpindicular bisectors are the set of points which are equidistant from two points. Therefore, their intersection is the point which is equidistant from all three points.
So, we have found the origin of our circle. Two observations are worth noting. First is that the perpindicular bisector of A and C is y=4. Since A and C are predetermined, O must lie on that line. Also observe that the farther that O is to the left, the greater the angle AOC becomes.
Now we create a circle with a radius of the distance between A and O and centered at O.
Now that we have A, B, and C on a circle, we can use the inscribed angle theorem. Recall that this theorem states that angle ABC= 1/2angle AOC. Therefore, if we maximize angle AOC, we also maximize angle ABC. Angle AOC increases as O moves to the left on the y=4 line. The origin is on the line y=4, so it is always a distance of 4 from the x-axis. This means that the radius of the circle can not decrease below 4 or our viewer is not lieing on the ground!
So, our new question is, far can we move the origin of our circle to the left, before the radius decreases to below 4... Well, we know that the distance from the origin to A is the radius of our circle. The origin's coordinates are (x, 4) where x is unkown and A's coordinates are (0, 2). According to the distance formula, this length is
. Setting this equal to 4 gives us that x=
. This gives us the origin point which maximizes the angle AOC, (
We see that we have maximized our angle AOC, so according to the inscribed angle theorem, we have also maximized our angle ABC. Therefor,e our viewer's head should be positioned
