Kickoff Returns and Geometry

After destroying Jaypreet, Jake, and Karmvir this fantasy football season (...not really), I began to wonder how far you could potentially return a kickoff while running in a straight line. While stats would only say a maximum of a 110 yard return, the Pythagorean Theorem shows that realistically, while running in a straight line, the distance between the back corner of one End Zone to the front corner of the other End Zone is approximately 122 yards. (TF2)

Measuring Height Two Ways

I decided it would be neat if I were to use similar triangles to find the height of the pole. I used a tape measure to measure out 20 feet from the pole and put a marker there. Then I took a picture with my phone.  I uploaded 2 copies of the picture to GSP and made two triangles.

They are both similar but have different scales; one is real world, and the other is on GSP. GSP gave me the lengths of the triangle in centimeters, so then all I did was set up a proportion to get the length of the pole. I also had a second way to do this: I set up a tangent equation to find the pole. Both results were very close but i think that the tangent equation is more accurate. (~TR1)

Sine with Roads

I found a picture to represent how sine could be used to find a real life distance. I also used Google Maps to find the distance of one of the legs of the triangle (EF). Since this is 0.9 miles and the sine of 29.85 degrees is about .49, the distance of DE is about .44 miles. I was also able to find the length of the hypotenuse using the Pythagorean Theorem and it came out to be approximately 1 mile. (~AL1)

Using Sine in Real Life

I created this triangle to use sine, cosine, or tangent to figure out a missing length. When I measured the angles it turned out that this triangle was a 30, 60, 90 triangle. The shelf leg forms an approximate 90 degree angle with the ground. Angle G is 60 degrees, and Angle E is 30 degrees.

Given: Angle of Depression = 60 degrees

            From the figure to the base of the shelf is 60 cm

Find: Height of shelf (AB)

By using the given information we can figure out a lot of things.

1. Angle C = 60 degrees

2. AC = 120 cm by using 30, 60, 90 triangle rules

We could find AB by using the 30-60-90 way, but let's use sine. Let's say AB, the height of the shelf, is x.

Sine=opposite/hypotenuse

sin(60 degrees) = x/120

120*sin(60) = x

x is approximately 104cm

If you check using the 30, 60, 90 way this is how it would be:

BC= 60

AB= 60√(3)

AB= 102

102 is approximately 104, check!

~SV1