Perimeter and Area in

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Perimeter and Area in the Coordinate Plane
9-4
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
Holt McDougal Geometry
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Warm Up Use the slope formula to determine the
slope of each line. 1. 2. 3. Simplify
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Objective
Find the perimeters and areas of figures in a
coordinate plane.
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In Lesson 9-3, you estimated the area
of irregular shapes by drawing composite figures
that approximated the irregular shapes 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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Example 1A Estimating Areas of Irregular Shapes
in the Coordinate Plane
Estimate the area of the irregular shape.
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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
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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Example 2 Finding Perimeter and Area in the
Coordinate Plane
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.
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
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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Check It Out! Example 2
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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Check It Out! Example 2 Continued
The opposite sides are parallel, so HJKL is a
parallelogram.
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Check It Out! Example 2 Continued
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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Check It Out! Example 2 Continued
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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Example 3 Finding Areas in the Coordinate Plane
by Subtracting
Find the area of the polygon with vertices
A(4, 1), B(2, 4), C(4, 1), and D(2, 2).
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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Check It Out! Example 3
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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Check It Out! Example 3 Continued
Area of triangles
b
a
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 are rearranged.
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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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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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Example 4 Continued
Find the area of each shape.
Left figure
Right figure
top triangle
top triangle
top rectangle
top rectangle
A bh 2(5) 10 units2
A bh 2(5) 10 units2
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Example 4 Continued
Find the area of each shape.
Left figure
Right figure
bottom triangle
bottom triangle
bottom rectangle
bottom rectangle
A bh 3(4) 12 units2
A bh 3(4) 12 units2
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Example 4 Continued
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
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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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Check It Out! Example 4
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.
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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.