9-4

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9-4 Perimeter and Area inthe Coordinate Plane9-4 Perimeter and Area in

the Coordinate Plane

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Warm Up

Lesson Presentation

Lesson Quiz

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9-4 Perimeter and Area inthe Coordinate Plane

Warm UpUse the slope formula to determine the slope of each line.

1.

2.

3. Simplify

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Students will be able to:Find the perimeters and areas of figures in a coordinate plane.

Learning Targets

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In Lesson 9-3, you estimated the area ofirregular shapes by drawing composite figures that approximated the irregularshapes and by using area formulas.

Another method of estimating area is to use a grid and count the squares on the grid.

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Estimate the area of the irregular shape.

Example 1A: Estimating Areas of Irregular Shapes in the Coordinate Plane

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Example 1A Continued

Method 1: Draw a composite figure that approximates the irregular shape and find the area of the composite figure.

The area is approximately 4 + 5.5 + 2 + 3 + 3 + 4 + 1.5 + 1 + 6 = 30 units2.

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Example 1A Continued

Method 2: Count the number of squares inside the figure, estimating half squares. Use a for a whole square and a for a half square.

There are approximately 24 whole squares and 14 half squares, so the area is about

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Check It Out! Example 1

Estimate the area of the irregular shape.

There are approximately 33 whole squares and 9 half squares, so the area is about 38 units2.

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Remember!

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Draw and classify the polygon with vertices E(–1, –1), F(2, –2), G(–1, –4), and H(–4, –3). Find the perimeter and area of the polygon.

Example 2: Finding Perimeter and Area in the Coordinate Plane

Step 1 Draw the polygon.

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Example 2 Continued

Step 2 EFGH appears to be a parallelogram. To verify this, use slopes to show that opposite sides are parallel.

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Example 2 Continued

slope of EF =

slope of FG =

slope of GH =

slope of HE =The opposite sides are parallel, so EFGH is a parallelogram.

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Example 2 Continued

Step 3 Since EFGH is a parallelogram, EF = GH, and FG = HE.

Use the Distance Formula to find each side length.

perimeter of EFGH:

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Example 2 Continued

To find the area of EFGH, draw a line to divide EFGH into two triangles. The base and height of each triangle is 3. The area of each triangle is

The area of EFGH is 2(4.5) = 9 units2.

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Draw and classify the polygon with vertices H(–3, 4), J(2, 6), K(2, 1), and L(–3, –1). Find the perimeter and area of the polygon.

Step 1 Draw the polygon.

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Check It Out! Example 2 Continued

Step 2 HJKL appears to be a parallelogram. To verify this, use slopes to show that opposite sides are parallel.

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are vertical lines.

The opposite sides are parallel, so HJKL is a parallelogram.

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Step 3 Since HJKL is a parallelogram, HJ = KL, and JK = LH.

Use the Distance Formula to find each side length.

perimeter of EFGH:


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To find the area of HJKL, draw a line to divide HJKL into two triangles. The base and height of each triangle is 3. The area of each triangle is

The area of HJKL is 2(12.5) = 25 units2.

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Find the area of the polygon with vertices A(–4, 1), B(2, 4), C(4, 1), and D(–2, –2).

Example 3: Finding Areas in the Coordinate Plane by Subtracting

Draw the polygon and close it in a rectangle.

Area of rectangle:

A = bh = 8(6)= 48 units2.

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Example 3 Continued

Area of triangles:

The area of the polygon is 48 – 9 – 3 – 9 – 3 = 24 units2.

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Find the area of the polygon with vertices K(–2, 4), L(6, –2), M(4, –4), and N(–6, –2).

Draw the polygon and close it in a rectangle.

Area of rectangle:

A = bh = 12(8)= 96 units2.

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Area of triangles:

a b

d c

The area of the polygon is 96 – 12 – 24 – 2 – 10 = 48 units2.

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Example 4: Problem Solving Application

Show that the area does not change when the pieces arerearranged.

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1 Understand the Problem

Example 4 Continued

The parts of the puzzle appear to form two trapezoids with the same bases and height that contain the same shapes, but one appears to have an area that is larger by one square unit.

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2 Make a Plan

Example 4 Continued

Find the areas of the shapes that make up each figure. If the corresponding areas are the same, then both figures have the same area by the Area Addition Postulate. To explain why the area appears to increase, consider the assumptions being made about the figure. Each figure is assumed to be a trapezoid with bases of 2 and 4 units and a height of 9 units. Both figures are divided into several smaller shapes.

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Solve3

Example 4 Continued

Find the area of each shape.

top triangle: top triangle:

top rectangle: top rectangle:

A = bh = 2(5) = 10 units2

Left figure Right figure

A = bh = 2(5) = 10 units2

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bottom triangle: bottom triangle:

bottom rectangle: bottom rectangle:

A = bh = 3(4) = 12 units2 A = bh = 3(4) = 12 units2

Example 4 Continued

Solve3

Find the area of each shape.

Left figure Right figure

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The areas are the same. Both figures have an area of

2.5 + 10 + 2 + 12 + = 26.5 units2.

If the figures were trapezoids, their areas would be

Example 4 Continued

Solve3

A = (2 + 4)(9) = 27 units2. By the Area Addition

Postulate, the area is only 26.5 units2, so the figures must not be trapezoids. Each figure is a pentagon whose shape is very close to a trapezoid.

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Look Back4

Example 4 Continued

The slope of the hypotenuse of the smaller triangle is 4. The slope of the hypotenuse of the larger triangle is 5. Since the slopes are unequal, the hypotenuses do not form a straight line. This means the overall shapes are not trapezoids.

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Create a figure and divide it into pieces so that the area of the figure appears to increase when the pieces are rearranged.

Check the students' work.

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Lesson Quiz: Part I

1. Estimate the area of the irregular shape.

25.5 units2

2. Draw and classify the polygon with vertices L(–2, 1), M(–2, 3), N(0, 3), and P(1, 0). Find the perimeter and area of the polygon.

Kite; P = 4 + 2√10 units; A = 6 units2

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Lesson Quiz: Part II

3. Find the area of the polygon with vertices S(–1, –1), T(–2, 1), V(3, 2), and W(2, –2).

A = 12 units2

4. Show that the two composite figures cover the same area.

For both figures, A = 3 + 1 + 2 = 6 units2.

9-4

Documents

area of efgh

area formulas

perimeter of efgh

method of estimating

area ofirregular shapes

coordinate plane step

learning targetsholt

half squares