GSE Analytical GeometryGeometric and Algebraic Connections

CirclesA circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point(the center) in the plane.

Can you find the radius of this circle?

If we were to place the circle on a coordinate plane we could use the Pythagorean Theorem tofind r:

Notice this is also the distance formula. The equation of a circle comes from the distance formula.

Standard Form of the Equation of a CircleThe standard form of the equation of a circle with center (h, k) and radius r is:

(x ­ h)2 + (y ­ k)2 = r2

Example 1:

Let's find the equation of a circle having radius 2 and center (­2, 1). Using the standard formgiven, substitute the value of r=2, h=­2, and k=1 into the equation and simplify it.

(x ­ h)2 + (y ­ k)2 = r2

(x ­ (­2))2 + (y ­ 1)2 = 22

(x + 2)2 + (y ­ 1)2 = 4

Now let's graph the circle. To graph the circle, use the center and graph it on the xy plane.

Since the radius is 2 (remember 22 = 4), you can plot points that are two above, two below, two tothe right, and two to the left of your center.

These are the points (­2, 3), (­2, ­1), (0, 1), and (­4,1). See the picture below:

The general form of an equation of a circle is given by whereA, B, and C are not all zero. For a circle, A equals B.

Therefore, they have the same value and sign.

Example 2:

Find the radius and center of 4x2 + 4y2 + 16x + 32y ­ 160 = 0.

To do this, we need to rearrange our variables and complete the square for both the x and the yportions of the equation.

4x² + 16x + 4y² + 32y – 160 = 0

Notice that each term is divisible by 4 so we can factor that out first.

x2 + y2 + 4x + 8y ­ 40 = 0

x2 + 4x + _______ + y2 + 8y + ________ = 40

x2 + 4x + 4 + y2 + 8y + 16 = 40 + 4 + 16

(x + 2)2 + (y + 4)2 = 60

The center is (­2, ­4) and the radius of the circle is which simplifies to .

To graph this circle you estimate the radius to be 7.75 and the graph appears as

Circles QuizIt is now time to complete the "Circles Quiz" quiz. You will have a limited amount of time;please plan accordingly.

Converting Standard Form to General Form Assignment

Select the "Converting Standard Form to General Form Assignment" Handout fromthe sidebar. Record your answers in a separate document. Submit your completedassignment when finished.

Using Shapes and Measures to Describe ObjectsGeometric shapes can be found in objects we use every day. Consider your current environment.Look around. What is the first object you see? What geometric shapes can you see within theobject? Look at the following images. Breakdown the images and name all the shapes that yousee. Rollover each image to see how they can be described.

Looking at objects as geometric shapes helps us describe them and use their properties formeasurement.

Using Density in Real­Life SituationsJust like using properties of geometric shapes to answer questions about objects around us, wecan use concepts of density based on area and volume to answer other real world questions.

Example

A big theater in Chicago has 1.5 million square feet. The manager of the theater surveyed thecrowd at the start of the last show to be sure it was not overcrowded. He found the populationdensity to be .011 people per square foot.

Given this information, how many people attended the show that day?

Answer:

To find this we need to consider the population density. This is equal to the number of people persquare feet of the theater.

Population Density = (number of people/square feet)

Now input the numbers given in the problem. We do not know the number of people in attendancethat day so we will use x to represent that value:

.011 = (x/1,500,000 square feet)

(.011)(1,500,000) = x

x = 16,500 people

16,500 people attended the theater that day.

Self­Check1. A car showroom has .76 million square feet. As the manager prepared for the fall special sale,he surveyed the showroom to be sure there was adequate space. He found the population densityto be .023 people per square foot. How many people were in the showroom? Solution

2. Some experts believe that elementary age children carry school bags that are too heavy. Theyare proposing that school bags not weigh more than 20 kg and have a volume of 15,000 cm3.Mackenzie's father measures her bag. The base of her bag is 11 cm by 15 cm and the height is 26cm. Does her Mackenzie's bag qualify if the density is .003 kg/cm3? (Use density = mass/volume)Solution

3. Carry­on luggage restrictions on most airlines have gotten stricter over the years. Currently,United Airlines requires carry­on baggage have a maximum volume of 27,000 cm3 and maximumweight of 10 kg. (Use density = mass/volume) Solution

View the following video to see how land productivity affects human population density.

Using Geometry in Design ProblemsNow that we have explored geometric shapes around us and used volume and density in real­lifesituations we can put the two together to answer design problems.

Consider opening a candy business. You want to sell candy bars at different sporting events. Thecandy bars you decide to sell come in boxes that are 1 foot long, 1 foot wide, and 3 feet high. Youexpect to do a lot of business so you will order several boxes of candy bars and store them in yourcloset. The dimensions of your closet are 8 feet long, 6 feet wide, 20 feet high. How many boxesof candy bars can you store in your closet?

Start by finding the cubic footage of your closet:

Area = (8 ft) (6 ft) (20 ft) = 960 ft3

Then find the area for a box of candy bars:

Area (1 ft) (1 ft) (3 ft) = 3 ft3

Finally, see how many boxes you can fit in your closet:

= 320 boxes

You will have space in your closet to store 320 boxes of candy bars.

Geometric Methods DiscussionYou are the owner of a local ice cream parlor. You have created a new ice cream treat to sell toyour customers. It is a cone filled with ice cream that will be pre­packaged. You would like thewrapper for this new treat to be a cone shape with a circular disk top. To save on production costyou need to determine the smallest amount of paper that would be needed to wrap the ice creamcone treats. Use the discussion board to post your ideas. Your initial post should:

1. Include a detailed description of the size and shape of the wrapper. (Try usingmeasurements for a real cone to help you come up with the actual size.)

2. Develop an equation to be used to calculate the surface area of the wrapper including thedisk top.

3. Pick dimensions of the radius of your cone base and slant height to then find the surfacearea of your cone.

4. Determine the rectangular dimensions for a sheet of paper measured in centimeters to thatwill be large enough to create your ice cream treat wrapper.

Module Wrap UpAssignment ChecklistIn this module you were responsible for completing thefollowing assignments.

Converting Standard Form to General FormAssignmentCircles and Parabolas QuizGeometric Methods DiscussionDr. Cone's New House ProjectGeometric and Algebraic Connections Test

ReviewNow that you have completed the initial assessments for this

module, review the lesson material with the practice activities and extra resources. Then,continue to the next page for your final assessment instructions.

Standardized Test PreparationThe following problems will allow you to apply what you have learned in this module to howyou may see questions asked on a standardized test. Please follow the directions closely.Remember that you may have to use prior knowledge from previous units in order to

answer the question correctly. If you have any questions or concerns, please contact yourinstructor.

Final AssessmentsFinal Module Test

It is now time to complete the "Geometric and Algebraic Connections" Test. Once youhave completed all self­assessments, assignments, and the review items and feelconfident in your understanding of this material, you may begin. You will have a limited

amount of time to complete your test and once you begin, you will not be allowed to restart yourtest. Please plan accordingly.

Dr. Cone's New House Project

A local mathematician, Dr. Cone, has hired your architecture firm to design his new house. Because your boss knows you are in Analytic Geometry, he has put you in charge of thedesign for the entrance of the house.

Please download the Dr. Cone New House Project from sidebar. Submit your completed projectwhen finished. A rubric is available in the sidebar.