Geometry in Action

 

My dad recently asked me what would be larger: a circle in which you increase the circumference by 2, or a circle in which you increase the radius by two. Using a few formulas (C=2πr; A=πr^2), it became apparent that the circle in which you increase the radius by two is considerably larger. I was able to illustrate this in GSP so you could get a dramatic visual of the question. (TF-2)

Circles in Mega Man II

Whilst playing Mega Man II, I unfortunately died. More or less, this is how I looked right before I did die:

 

All silliness aside, I did notice that Mega Man explodes into a bunch of circles upon death.

The circles Mega Man explodes into when he dies share all of the qualities of circles and can be used to demonstrate them:

For example, these circles can be used to demonstrate Theorem 9-11, which states “when two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.” As shown by my diagram, AX * BX =CX * DX, which is exactly what the theorem states. (~JC1)

And no, Channing, Air Man (pictured below) is not wearing some kind of fancy pants.

Pizza with Similar Triangles

This is a picture of a pizza slice wrapped in aluminum foil. Although it may be hard to see, the foil is folded in such a way that the edge is parallel to the opposite edge of the pizza, therefore forming two triangles. These two triangles are similar.

Guide:
A=point to the lower left
B=point farthest to the top
C=point farthest to the right
D=point on line AB
E=point on line AC

Proof:

Statements:                                  Reasons:
1. m<A=m<A                              1. Reflexive Property
2. Line DE and line BC are ||.      2. Given
3. m<ADE=m<ABC                    3. If two parallel lines are cut by a transversal, then corresponding                                                          angles are congruent.
4. Triangle ABC is congruent     4. AA Postulate
to triangle ADE.

Instead of using <ADE and <ABC, I also could have used <AED and <ACB. Both would fulfill the criteria of using the AA Postulate, that two angles from each triangle must be congruent for the triangles to be proved similar. (~RG1)

Great Tasting Geometry

Over the weekend, I went to my favorite restaurant in Santa Cruz called Tramonti's. I ordered my favorite pizza with ham, mushrooms, and a huge egg on top, and I couldn't resist making a collage of it to show geometric similarities! On the far right of the picture, I have half of the pizza, or a semi-circle equaling 180 degrees. I showed the Arc Addition Postulate because arc ADB plus arc BED is equal to the total of arc ABC. In the top left hand corner of the picture, I proved Theorem 9-3. I showed that the minor arcs FG and HG were congruent because their central angles were congruent. I also was able to prove Theorem 9-5 in the same picture. Diameter FH is perpendicular to chord JK so it bisected the chord and its arc. In the bottom left hand corner, I am just showing that the pizza slice made an isosceles triangle with MN congruent to OM. This pizza was not only delicious, it was full of geometry! (~MG1)

Parallel Yardlines

ABC = 155

DCB = 25

This is an example of parallel lines by Theorem 3-6, which says that if two lines are cut by a transversal and the same-side interior angles are supplementary then the lines are parallel. (~CB)

(CB per 5)

Regular Octogan

When I was at a hotel recently, I noticed that the table was a regular octagon. Because this picture was taken at a distance, the table is distorted. From straight on top the table would look more like this.

I wanted to find the area of the octagon.  The equation for this is (1/2)(apothem)(perimeter).  The apothem is the length from the center to the midpoint of one of the sides.

Here, segment OI is the apothem and is 4.35 cm.  Each side is 3.61 cm. The real table's sides were about 0.6m or 60cm. To find the real apothem we can set up a proportion with
                                                                               

                           







This equals 72.299cm, the length of the apothem.  Then, we can find the area of the octogon

(~NW5)

Pi(e) Day

On Pi Day, I was very disappointed when we didn't eat pie. But then we started to talk about finding circumferences using pi, and in that moment I knew that was my mission...or at least an idea for a post. Using a real pie, I can demonstrate the function of pi to find the circumference of said pie and see if the ratio is correct. (~CB1)

Externally Tangent M&M's

A common external tangent is a tangent that does not intersect the segment joining the centers.

When I saw my sister arranging her M&M's on the table, I immediately noticed a pattern of externally tangent circles, which are coplanar circles that are tangent to the same line (just imagine it) and point. The M&M pattern shows externally coplanar circles -- the yellow points are the points of tangency and the tangent line that the circles intersect on. And because the circles are the same radius, the six M&M's around the inner circle fit perfectly. (~IR1)

iPhone Camera Externally Tangent

This is a picture of the camera and flashlight on an iPhone. The two circles of the flashlight and the camera are externally tangent. Tangent circles are coplanar circles that are tangent ot the same line at the same point. (~RG1)

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)

Parallel or Not?

These are bricks arranged in a pattern to create the back walls of a fireplace. I wanted to see if the lines were exactly parallel. However, the alternate interior angles are not congruent, meaning that according to my GSP diagram, the edges of the brick are not congruent. Also notice that to make this pattern, the bricks have to be exactly 2x longer than they are wide... (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)

Dining Table

This is a picture of my dining table. I wanted to see how parallel the different planks were on the table. I decided to use the corresponding angle postulate and I got these angles. They aren't quite parallel but are very close. (~TR1)

Topology

To a regular person, these objects don't look similar. But according to the science of topology, a modern area of geometry which studies all different types of spaces, these objects belong in the same group.

EOS Chapstick

 

Nutella®  Jar 

When two figures belong in the same group, they are referred to as topologically equivalent. The objects of study are most commonly organized by figures that can be stretched, bent or molded into different shapes without cutting or puncturing, and vice versa.

So, by those standards, the EOS and Nutella® would fit in the same category, because they have both have a shape that can be changed to the other without cutting puncturing. (~IR1)

Pentagons Within a Lotus Chandelier

I saw this giant lotus chandelier, probably about 20 feet tall, in a shop in Chinatown, San Francisco.

From the naked eye I thought I found a regular pentagon within the body of the chandelier, but I was rather disappointed to find that it is only a pentagon (not regular). All the side lengths are different, but the angles add up to 540 degrees. Since the differences in length are so minute, I can see why I assumed it a regular pentagon.  (~LU2)

Triangle Within a Tree

Within the fan-shaped tree branches, an obtuse triangle is formed with the help of the rooftop of Baskin Robbins.(~LU2)

Stylish Geometry

I went to the Gilroy Outlets last weekend and visited the Coach store.  I was looking at the purses when I realized that some of the purses were perfect isosceles trapezoids.  I used Geometer's Sketchpad to prove Theorem 5-18 and Theorem 5-19.

Theorem 5-18 states that the base angles of an isosceles trapezoid are congruent and my picture shows that the base angles are both 69.89 degrees. In Theorem 5-19, the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths.  Since segment AD and segment BC were already parallel, then segment EF was parallel as well.  I then measured and added segments AD and BC and concluded that the average was 8.71, which is the same as EF.  Therefore, I  proved that this Coach purse is an isosceles trapezoid and a fashionable accessory!  (~MG1)

Chocolates!

This is a picture of a Nestle chocolate tin container I saw at my friend's house. This demonstrates the theorem stating, "The sum of the measures of the angles of a convex polygon with n sides is (n-2)180." The chocolate tin was in the shape of an octagon, which obviously has eight sides. So the sum of the measures of the interior angles of this polygon is (8-6)180=1080 degrees. Each individual angle was 135 degrees, and 135(8)=1080. Yay! (~RG1)

The Television: Two Rectangles

This is picture of my TV at home. There is the screen, the first rectangle, and the frame, the second rectangle. In real life, the four pairs of sides of the rectangles are parallel. Also, one characteristic of rectangles is that all of their included angles are right angles, as shown in the image. (~RG1)

Vertical Angles

AGE = 97

CGB = 97

DHE = 94

BHF = 94

These are examples of vertical angles. In geometry you learn that all vertical angles are congruent. From this picture you can see that angle AGE=CGB. (~CB5)

Similar Triangles with Parallel Sides

 

While listening to music, I noticed that the logo on my headphones is an example of two similar triangles. To prove this, I traced both triangles, and then found the midpoints of lines AC and BC. These two points happened to land precisely where expected, and so I connected them to form a white overlay of the symbol.

I used the measuring tool in GSP to find all five angles, which are listed on the left. By using the AA postulate, we can see that triangle ABC and triangle EDC are indeed similar.

Although it's not included in the picture above, I also measured angle AED. The sum of angles AED and BAC was 180 degrees, which proves that side AB is parallel to side ED.

(~GC2)

Congruent Squares

These squares are congruent.  You can tell because they have parallel lines which are cut by a transversal. It follows that the interior angles are congruent.  It also shows that the same-side interior angles are supplementary.  This is obvious to see because a square has four 90 degree angles.  The squares also share a side and the same parallel lines. In conclusion, these two squares are congruent. (~KM5)

Carpeted Rectangles

Earlier today, I was looking at the carpet on my living room floor and I saw what appeared to be three similar quadrilaterals, more specifically rectangles.

I then took a picture of it and uploaded it to sketchpad to prove Theorem 5-12, which states that the diagonals of a rectangle are congruent. I measured the lengths and proved that it was a rectangle and then proved that the diagonals were congruent.

(~WM5)

Mario's Parallel Coins

While playing a computer version of the classic Super Mario Bros. that I found online, I found a secret underground area, which is pictured above. In the underground area, I noticed that the coins seemed to be perfectly parallel to each other.

And, as it turned out, I was correct. By placing a point at the center of each coin and drawing lines between them, the coins formed lines parallel vertical lines and parallel horizontal lines. One can prove them parallel because the corresponding angles are congruent, the alternate interior angles are congruent, and the same-side interior angles are congruent. Because all of the angles formed by the intersections of the vertical and horizontal lines all have a measure of 90 degrees, they are perpendicular. This shows something important about coplanar horizontal and vertical lines: they are always perpendicular to each other. (~JC1)

Quilting Stars

My grandmother was a quilter, and quilts need geometric designs to fit together well. In this quilt, there are beautiful stars. Stars consist of both concave and convex angles. These stars wouldn't be considered "regular" because they are uneven in length and angle measure. The exterior angle measure of these stars would not be 360 like a regular polygon (it is actually 401.48), because of the concave and convex angles, but they sure are pretty! (~CB1)

Divergent. Yay.

This is my drawing of Eric from Divergent by Veronica Roth. I used a grid because I based it on another picture. I gridded the other picture, which was this guy (Gerard Way from My Chemical Romance):

My drawing also contains many sets of parallel and perpendicular lines.

(~CS1)

Geometric Window

This is a picture I took of my garage window. I noticed more than one geometry concept in it. It is an example of parallel lines, congruent angles, and perpendicular lines. I compared the lengths of each segment with the length of my phone and they are indeed equal. -(CD1)

Geometric Sydney

While I was visiting Australia over February break with my mom, I took a head on picture of the Sydney Opera House from a ferry I was on.  From this picture, you can see the demonstration of Theorem 4-7, which states - "If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.  The solid green line down the center is the angle bisector, as the other two green lines on the sides are the sides of the angle, which is 91.84 degrees.  It then split the angle into two smaller angles of 45.92 degrees each.  I then constructed a point on the line and connected it to the two sides, showing that they are each the same distance, 2.39 cm. (~JG2)

Triangle Tiles

These triangular tiles are displaying congruency and vertical angles. The triangles are isosceles because the two sides are congruent; the opposite angles are also congruent (Theorem 4-1). The triangles are all congruent by the SSS Postulate, because all three sides are congruent to three sides of another triangle. (~AV5)

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)