Hamster Wheel

I was feeding my hamster and noticed the wheel that she runs on. On the wheel are two steel segments that are running through the circle. These segments are both congruent and are both the diameter of the wheel. Since the chords of the circle (the diameter) are congruent this displays Theorem 9-4, which is that congruent chords have congruent arcs. Both the arcs share the same central angle which also prove that the arcs would be congruent. (~AV5)

Baskets and Circles

I saw this basket that was holding oranges in my house and realized that it had an interesting design on it. The diameter is splitting the circle in half creating two congruent arcs. The other two chords in the circle share the same central angle. This means that the arcs would be congruent based off theorem 9-3, which is that if arcs share central angles then they are congruent. the chords would also be congruent because congruent arcs have congruent chords. (~KM5)

Congruent Segments and Arcs

This is a semi-circle window that is in the front of my house. In this window the 3 segments CF, DF, and BF are congruent through the one central angle to the edge of the arc. Also in this picture the 4  angles coming out of the center point F are congruent to each other because of their congruent arcs and central angle. Note: the measurements are not exactly congruent because the point is not in the exact center of the window. (~AD1)

Bent Shower Rod

This past week I went to Monterrey with my mom for a little vacation. We stayed in a small hotel and while there I noticed the bent shape of the shower rod. I immediately thought of geometry.

After drawing in a straight bar and calculating the area of the triangle formed versus the increased area with a bent shower rod:

I concluded that the bent rod greatly increases the space within the shower.
37.31-21.63= 15.68 cm-squared more space.

This is probably one of the simplest yet one of the most useful inventions for everyday life. (~LU2)

Angles and Arcs in Clocks

In this picture angle EJF equals 135.15 and is congruent to its arc EF which equals 135.19. This is because a central angle inscribed in a circle is congruent to its arc.

[Note: The numbers are slightly off though because the angle is not exactly in the center of the clock. This makes the numbers not perfectly equal but if the angle was in the center the angle and the arc would be congruent.]

 (~AD1)

Proving Theorem 9-5 with a Car Tire

As theorem 9-5 states, "a diameter that is perpendicular to a chord bisects the chord and its arc." I decided to use a picture of my car tire to show the proof of this theorem. 

~ (JC2)

Arcs and Central Angles

This picture is of a CD disk in its cover. This picture shows two different ideas. One idea is a quadrilateral circumscribed about a circle. The cover fits around the disk like a quadrilateral circumscribed about a circle. The second idea is arcs and central angles. Point A is the center point here. Arc BC is approximately 37 degrees. The central angle CAB is also 37 degrees.

Angles on a Clock

When I was looking at my clock I noticed that not only could I find the angles made by radii of the triangles but also secants using the theorems in chapter 9. By theorem 9-4 I determined that the measure of angle BCD was equal to the measure of arc BD. From that I was able to use theorem 9-7 to prove the measure of angle BAD is equal to half of arc BD. Therefore the measure of angle BAD is equal to half of the measure of  angle BCD. (~AL1)

GIRL

At my local swimming pool, I found a women's bathroom sign with the shape of a perfect circle and was wondering if I could prove any geometric theorems with it.

On Geometer's Sketchpad, I drew a tangent of the circle and a secant to the circle. I planned to prove theorem 9-10 which states that two secants, two tangents, or a secant and a tangent follow the same equation. The angle that is formed by the intersecting lines would be equivalent to 1/2 the quantity of the major arch created minus the minor arch created.

Through GSP, I have proven that no matter what size or angle, the equation works every time.

(~WM5)

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)