holt geometry 9-4 perimeter and area in the coordinate plane 9-4 perimeter and area in the...
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Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane9-4 Perimeter and Area in
the Coordinate Plane
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Warm UpUse the slope formula to determine the slope of each line.
1.
2.
3. Simplify
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Find the perimeters and areas of figures in a coordinate plane.
Objective
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
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.
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Estimate the area of the irregular shape.
Example 1A: Estimating Areas of Irregular Shapes in the Coordinate Plane
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Remember!
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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:
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Check It Out! Example 2 Continued
are vertical lines.
The opposite sides are parallel, so HJKL is a parallelogram.
Holt Geometry
9-4 Perimeter and Area inthe 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
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Example 3 Continued
Area of triangles:
The area of the polygon is 48 – 9 – 3 – 9 – 3 = 24 units2.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe Coordinate Plane
Example 4: Problem Solving Application
Show that the area does not change when the pieces arerearranged.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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
Holt Geometry
9-4 Perimeter and Area inthe 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
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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.
Holt Geometry
9-4 Perimeter and Area inthe 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
Holt Geometry
9-4 Perimeter and Area inthe 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.