This document contains a geometry lesson on calculating the perimeters and areas of polygons in the coordinate plane. It includes examples of estimating areas of irregular shapes using grids, finding perimeters and areas of parallelograms, and subtracting areas of additional shapes from a bounding rectangle. One example shows two composite figures that appear to have different areas but are proved to have the same area through calculation. The lesson concludes with a quiz to assess understanding of finding perimeters and areas of polygons in the coordinate plane.

1. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane9-4
Perimeter and Area in
the Coordinate Plane
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz

2. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Warm Up
Use the slope formula to determine the
slope of each line.
1.
2.
3. Simplify

3. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Find the perimeters and areas of
figures in a coordinate plane.
Objective

4. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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.

5. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Estimate the area of the irregular shape.
Example 1A: Estimating Areas of Irregular Shapes in
the Coordinate Plane

6. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

7. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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

8. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

9. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Remember!

10. Holt Geometry
9-4
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.
Example 2: Finding Perimeter and Area in the
Coordinate Plane
Step 1 Draw the polygon.

11. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Example 2 Continued
Step 2 EFGH appears to be a
parallelogram. To verify this,
use slopes to show that
opposite sides are parallel.

12. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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.

13. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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:

14. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

15. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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.

16. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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.

17. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Check It Out! Example 2 Continued
are vertical lines.
The opposite sides
are parallel, so HJKL
is a parallelogram.

18. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Step 3 Since HJKL is a parallelogram, HJ = KL,
and JK = LH.
Use the Distance Formula to find each side length.
perimeter of EFGH:
Check It Out! Example 2 Continued

19. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

20. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

21. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Example 3 Continued
Area of triangles:
The area of the polygon is 48 – 9 – 3 – 9 – 3 = 24 units2
.

22. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.

23. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Check It Out! Example 3 Continued
Area of triangles:
a b
d c
The area of the polygon is 96 – 12 – 24 – 2 – 10 =
48 units2
.

24. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Example 4: Problem Solving Application
Show that the area
does not change
when the pieces are
rearranged.

25. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
11 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.

26. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
22 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.

27. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Solve33
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

28. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
bottom triangle: bottom triangle:
bottom rectangle: bottom rectangle:
A = bh = 3(4) = 12 units2
A = bh = 3(4) = 12 units2
Example 4 Continued
Solve33
Find the area of each shape.
Left figure Right figure

29. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
Solve33
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.

30. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
Look Back44
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.

31. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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.

32. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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

33. Holt Geometry
9-4
Perimeter and Area in
the Coordinate Plane
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
.