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

The Interior Angle Equation of a Polygon

Here is an example of how the equation used to find the interior angle measure sum in polygons works.

Angle sum = (n-2)(180)

This is a Lamborghini that I photographed at a car show. Notice how the headlights form pentagons. Now we will test to see if the equation works.

To find the sum of the interior angles in a pentagon, the equation is (5-2)(180).

(5 - 2)(180)
= 3(180)
= 540

So, the total measure of all the angles in a pentagon equals 540 degrees. This equation should work for all pentagons, so now let's test to see if it works on the headlight.

As we can see, the measures of all the angles are: 54.56, 129.09, 103.2, 136.76, and 116.39. If we add these all together, we get 540, which proves the equation to be correct, for both regular and non-regular polygons. (~DM5)

Congruent Angles in a Stool

I noticed that a stool in my house had rungs that were parallel to each other. I figured that if the rungs were parallel then the corresponding angles would be congruent. After taking the picture and putting it into GSP, I found that angle BAF was indeed congruent to angle DCF. (~DM2)

Perfectly Perpendicular?

My family is currently renovating the kitchen, and the workers just put in the tile backsplash. Supposedly they have tools to make sure they place the tile perfectly, but I decided to see if it actually aligned. Surely if the tile was placed perfectly, then there should be a perpendicular line in between the tiles.

By definition, a perpendicular line is one that has a right angle (which is 90 degrees) on both sides. However, when I measured the angles of the tiles, it looks like the tile guys were a little off. One side is about 90.29 degrees and the other side is 89.71 degrees. It's pretty close but not perfect. Sorry guys but I just proved you wrong....It still looks really good though! (~JC2)

Isoceles Triangles

I was at swim practice the other day and I noticed that the backstroke flags were upside down isosceles triangles. Two of the sides of the flags are congruent and the angles opposite to those sides are congruent as well making it an isosceles triangle by the Isosceles Triangle Theorem. (~AD1)

Geometric Tiles

These tiles are an example of if two parallel lines are cut by a transversal, alternate interior angles are congruent. (~JZ5)

Tessellations in a Table

I saw a table and noticed it had tessellations in it. First off, the shapes are hexagons. This can be proved using the formula (n-2)180 where n is the number of sides and the sum should be the sum of the angles. So (6-2)180= 720 degrees.

The hexagons fit together at the intersections because the two smaller angles are about 50 degrees and the two larger angles are about 130 degrees.  Altogether it is 360 degrees.  This means there are no gaps in the tessellation, and it fits together well.  (~NW5)