Geometry Lab - Three Point Circle

Follow instructions for LABS -as outlined in the syllabus (about neatness, organization and the like)

We spoke in class how circle may be defined by either its center and its radius, a point and its center (the distance between them implies a radius) or alternatively by a system of three points that mark out the circle's edge. Three points may be used to find first the center and then the equation of the circle whose edge they are on by using a combination of the perpendicular bisectors of the chords and the distance formula. The following instructions will guide you through a process that will lead you to the circle I took your points from. As you go through this process you will have to print off certain results you get using the equation grapher program on green globs. There are a couple things you need to know.

GREEN GLOBS is installed on all the computers in the computer lab. When you double click on it where it is found in the upper left hand corner of the desk top, it will open the program which actually houses a number of other programs in the one package. The one you will need of for this assignment is the equation grapher. Open up that program by clicking programs and then clicking equation grapher. You might want to read first how to write equations before actually using it for your lab. You'll find "How to write equations" in the Equation Grapher menu along with the various windows sizes available for graphing equations on. You'll use (-10,-8) to (10,8)

To use it for your lab, you will type in and graph various equations that you have figured out by hand in the steps below. You will do this to verify that you are working through them properly. if you are not it will hopefully become apparent to you quickly.

There is a little problem with the new version of green globs in that it does not have a print screen command under file options and so you will have to perform the following actions to show me your results.

To print the results:
1) open a word documen and change the page orientation to portrait
2) type in your name and the date along with the title geometry lab
3) go to the green globs screen that you are trying to print and once its on your screen and ready
4) hit the Print Screen/SysRq key that is in the top row of the keyboard. This key takes a screen shot of your work which you can then paste [CTRL-V] into your open word document.
5)Go back to your word document and paste it in sizing it appropriately to fill the space. You can also mess with the zoom stuff before taking the screenshot but it shouldn't be necessary. If you keep pasting to the same document you can save it all as one file.

Okay, Now the lab. A lot of this you can do ahead without using the Green GLobs equation grapher tool. In fact you could do the whole thing and then go graph it, but if you made a mistake that mistake would not be revealed until you the end and that's never any fun.

THe Directions

1. In class you were given or choose three points (__, __) , (__, __), and (__, __). In the order that you chose them call them points A, B and C (as this allows me to simplify my directions) and on a graph paper you will include with your results and calculations mark those points and connect them to make the triangle ABC.

2. Using the midpoint formula, find the midpoint of each side of the triangle ( side AB, side BC, and side AC) showing the necessary calculations and adding each midpoint to your graph.

3. Find the slope and equation of the line containing each side of the triangle showing all those calculations.

3b. Use the green globs equation grapher to check your results graphed together make the correct triangle. Print out your result, write in the scale if not shown and labeling the points.

4. Using the previous two results write the equation of the perpendicularbisector for each side of the triangle. That is to say find the equation of the line through the midpoint that is perpendicular to each side of the triangle.

4b. Graph these along with your answers from 3b with the green globs grapher to demonstrate that they do cut each side in half at a right angle. (also if done correctly they should all meet in a single point). You should do this before you proceed further.

5. The point of intersection between the perpendicular bisectors of the triangle is called the circumcenter. Find it by using substitution or elimination method to identify the coordinates of the point of intersection for two of your perp. bisectors.

5b. Check your answer by plugging it into the remaining equation. If you have performed all parts correctly thus far, The resulting coordinate from the previous step should work in your remaining perp. bisector's equation. (you may also do an optional check using your graphing calculator by using 2nd trace (CALC) #5 intersection with two of the equations).

6. Almost done, we've found the center of the circle. Use the distance formula with the center and one of the points you started with to find the radius.

7. Write the equation of the circle using the radius you found and the circumcenter as the center. Check that each point substituted into the equation as a (x, y) validates your result.

Now Perform the following two graphical checks. I'd have you do both at once but I believe Green Globs only supports 6 equations at a time in the Equation graphing program.

7b. Using the Green Globs equation grapher, input all the equations for the sides again from question 3 then input your equation for the circle. If performed correctly. Your circle should cross through each corner of the triangle. Print this off and include it in your results.

7c. Again using the equation grapher, input the three equations you found for the perpendicular bisectors and graph them along with the equation of the circle you found. if you did this correctly the center of the circle should be the intersection of the equations.

8. On a new piece of graph paper, graph the triangle, the midpoints and center, connect the midpoints to and through the center in both directions for the perpendicular bisectors and when you com to class I'll loan you a compass to complete the circle you whose equation you found.

Turn in the whole packet to me at the time of the final.

Asssignment 19

Section 4-4 / pg 165-166 / special questions

For #21a & b / calculate how long it would take for the investment of $900 to reach a maturity value of $2000 at the rate given.

#25 How long for Sean's money to double?

#K1 In 1995 the price of a movey ticket was $6.50. It is now $9.50. Use the continuously compounding interest formula to calculate the rate of inflation.

#K2 Suppose in four years a CD that was compounded montyhly grew 23%. Find the annual rate.

#K3. Evaluate y = A(b)^(kx)
that is y is equal to A (some amount) times b (the base) raised to the k times x power.
a. When A = 30, b = e and k = .05 and x = 0
b. When A = 15,000, b = 1.08, k = 1 and x = 10
c. When A = 120,000 , b = e, k = -.012 and x = 30
d. Find x when A = 3, b = e, k = 2, y = 445.24
e. Find x when A = 180,000 , b = 3 , k = 2 and y = 20,000

#K4. One very important exponential equation is the compound-interest
formula. It says:

A = P ( 1 + r/n)^(nt)

...where "A" is the ending amount, "P" is the beginning amount (or "principal"), "r" is the interest rate (expressed as a decimal), "n" is the number of compoundings a year, and "t" is the total number of years. The formula calculates the amount (A) owed a person who leaves their money (P) in an account that compounds interest in n times per year, based on a rate of r % Interest per year, for t years of time.

a. Use this formula to calculate the amount of money $10,000 would earn if it were invested at 4.5 percent per year for 8 years in an account that compounded the interest monthly (12 times per year)

b. Use this formula to calculate how long until the investment will have earned a 50% return (that is how long until the investment earns $5000 in interest and is valued at $15,000 total)

c. Use this formula to calculate what interest rate double a person's initial investment in 12 years if the account the account compounded the interest on a monthly basis.