Area and Perimeter of Triangles on the Coordinate Plane

Looking at Something Familiar in a New Way

Area and Perimeter of Triangles on the Coordinate Plane

Learning Goals

In this lesson, you will:

? Determine the perimeter of triangles on the coordinate plane. ? Determine the area of triangles on the coordinate plane. ? Explore the effects that doubling the area has on the properties of a triangle.

3.2

? Carnegie Learning

One of the most famous stretches of ocean in the Atlantic is an area that stretches between the United States, Puerto Rico, and Bermuda known as the Bermuda Triangle. A heavily traveled area by planes and ships, it has become famous because of the many stories about ships and planes lost or destroyed as they moved through the Triangle. For years, the Bermuda Triangle was suspected of having mysterious, supernatural powers that fatally affected all who traveled through it. Others believe natural phenomena, such as human error and dangerous weather, are to blame for the incidents.

235

Problem 1 Determining the Area of a Triangle

1. The formula for calculating the area of a triangle can be determined from the formula for the area of a parallelogram.

A

D

B

C

a. Explain how the formula for the area of a triangle is derived using the given parallelogram.

3

b. Write the formula for the area of a triangle.

2. Graph triangle ABC with vertices A(27.5, 2), B(25.5, 13), and C(2.5, 2). Then, determine its perimeter.

y

16

12

8

4

28 26 24 22

02 4 6 8

x

24

28

212

216

? Carnegie Learning

236 Chapter 3 Perimeter and Area of Geometric Figures on the Coordinate Plane

3. Determine the area of triangle ABC. a. What information is needed about triangle ABC to determine its area?

?b. Arlo says that line segment AB can be used as the height. Trisha disagrees and says that line segment BC can be used as the height. Randy disagrees with both of them and says that none of the line segments that make up the triangle can be used as the height. Who is correct? Explain your reasoning.

3

c. Draw a line segment that represents the height of triangle ABC. Label the line segment BD. Then, determine the height of triangle ABC.

Do I use line segment AC or line segment BC as the length of the base?

d. Determine the area of triangle ABC.

? Carnegie Learning

3.2 Area and Perimeter of Triangles on the Coordinate Plane 237

4. Perform one or more transformations to help you determine the perimeter and the area more efficiently. a. Transform triangle ABC on the coordinate plane. Label the image A9B9C9. Describe the transformation(s) completed and explain your reasoning.

y

16

12

8

4

28 26 24 22

02 4 6 8

x

24

28

3

212

216

b. Determine the perimeter of triangle A9B9C9.

? Carnegie Learning

238 Chapter 3 Perimeter and Area of Geometric Figures on the Coordinate Plane

c. Determine the area of triangle A9B9C9. Be sure to label the height on the coordinate plane as line segment B9D9.

5. Compare the perimeters and the areas of triangles ABC and A9B9C9. a. What do you notice about the perimeters and the areas of both triangles?

3

b. Use what you know about transformations to explain why this occurs.

3.2 Area and Perimeter of Triangles on the Coordinate Plane 239

? Carnegie Learning

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