Maths

Task 1

Area of a Triangle

Finding the area of a triangle is straightforward if you know the length of the base and the height of the triangle. But is it possible to find the area of a triangle if you know only the coordinates of its vertices? In this task, you’ll find out. Consider ΔABC, whose vertices are A(2, 1), B(3, 3), and C(1, 6); let line segment AC represent the base of the triangle.

Part A

Find the equation of the line passing through B and perpendicular to.

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Part B

Let the point of intersection of  with the line you found in part A be point D. Find the coordinates of point D.

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Part C

Use the distance formula to find the length of the base and the height of ΔABC.

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Part D

Find the area of ΔABC.

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Part E

Now check your work by using the GeoGebra geometry tool to repeat parts A through D. Open and complete each step below. If you need help, follow these instructions for using GeoGebra. You will take a screenshot of your work when you are through, so be sure to clearly label your construction as you work.

1. Plot points A, B, and C, and draw a polygon, ΔABC, through the points.
2. Draw a line perpendicular to  through point B.
3. Label the intersection of the line perpendicular to  through B and  point D.
4. Measure and display the slopes of  and .
5. Display the equations of  and  in the Algebra margin.
6. Measure and display the lengths of  and .
7. Calculate and display the area of ΔABC.

Take a screenshot showing the geometric construction and the Algebra margin, save it, and insert the image in the space below.

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Part F

Compare the calculations displayed in GeoGebra with the calculations you completed in parts A through D. Look in the Algebra margin too. Do the results in GeoGebra match the results you obtained earlier? If not, where do the discrepancies occur? You might have to rearrange equations algebraically to determine whether two equations match.

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Part G

You’ve seen two methods for finding the area of ΔABC—using coordinate algebra (by hand) and using geometry software. How are the two methods similar? How are they different? Why might coordinate algebra be important in making and using geometry software?

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Task 2

Area of a Polygon

In this unit, you learned about finding the area of triangles and rectangles using coordinates. But there are many more shapes than just triangles and rectangles, and these shapes often can be relatively complex. You can still find properties of such polygons by dividing them into an arrangement of simpler shapes—polygons that you’re familiar with.

When a two-dimensional figure is composed of smaller figures, its area is equal to the sum of the areas of the smaller figures. When finding the area of a polygon, it is helpful to know that any polygon can be partitioned into a series of triangles.

You will use GeoGebra to partition a polygon into a set of triangles. Go to partitioning a polygon http://contentstore.ple.platoweb.com/content/GeoGebra.v5.0/B2_Polygon_Partitioning.html, and complete each step below.

Part A

Draw line segments to partition the given polygons into triangles. There is more than one correct answer. Take a screenshot of your construction, save it, and insert the image in the space below.

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Part B

Think back to Task 1 of these Lesson Activities. Finding the area of a polygon using only coordinates can be tedious. Fortunately, you do have access to additional tools that help you find the area of a polygon while following the same basic methods. This is especially helpful when polygons are irregular or have many sides.

Next you will use GeoGebra to find the area of a polygon divided into a set of triangles. Go to area of a polygon http://contentstore.ple.platoweb.com/content/GeoGebra.v5.0/B2_Area_of_a_Polygon.html , and complete each step below:

1. Partition the polygon into triangles by drawing line segments between vertices.
2. For each triangle, draw an altitude to represent the height of the triangle. Place a point at the intersection of the height and the base of each triangle.
3. Use the tools in GeoGebra to find the length of the base and the height of each triangle. (Because the values displayed by GeoGebra are rounded, your result will be approximate.)
4. Compute the area of each triangle, and record the results below. Show your work.
5. Add the areas of the triangles to determine the area of the original polygon, and note your answer below.

When you’re through, take a screenshot of your construction, save it, and insert the image below your answers.

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Part C

In part B, you used a combination of GeoGebra tools and manual calculations to find the approximate area of the original polygon. Now try using the more advanced area tools in GeoGebra to verify your answer in part B. Which method did you choose? Do your results in parts B and C match?

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Task 3

Parabolic Reflectors

In flashlights, the reflector that directs the light from the bulb is parabolic. If the center of the bulb is placed at the focus of the reflector, then all the generated light will be directed forward in a concentrated beam. In a parabolic reflector, taking a cross section of the reflector through its central axis will create a parabola as shown in the diagram.

Part A

A manufacturer is designing a flashlight. For the flashlight to emit a focused beam, the bulb needs to be on the central axis of the parabolic reflector, 3 centimeters from the vertex. Write an equation that models the parabola formed when a cross section is taken through the reflector’s central axis. Assume that the vertex of the parabola is at the origin in the xy coordinate plane and the parabola can open in any direction.

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Part B

Find the equation of the directrix of the parabola found in part A.

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Part C

Based on feedback from an independent research firm, the flashlight manufacturer has decided to change the design of the flashlight. The reflector now needs to extend 4 centimeters past the center of the bulb, as shown in the diagram. In the new design, how wide will the reflector (CD) be at its widest point? Show your work.

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