correctionkey=nl-d;ca-d name class date 10.5 perimeter and...

Use the Distance Formula to find the remaining side lengths.

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Explore Finding Perimeters of Figures on the Coordinate Plane

Recall that the perimeter of a polygon is the sum of the lengths of the polygon’s sides. You can use the Distance Formula to find perimeters of polygons in a coordinate plane.

Follow these steps to find the perimeter of a pentagon with vertices A (−1, 4) , B (4, 4) , C (3, −2) , D (−1, −4) , and E (−4, 1) . Round to the nearest tenth.

A Plot the points. Then use a straightedge to draw the pentagon that is determined by the points.

B Are there any sides for which you do not need to use the Distance

Formula? Explain, and give their length(s).

C

D Find the sum of the side lengths.

Reflect

1. Explain how you can find the perimeter of a rectangle to check that your answer is reasonable.

Module 10 549 Lesson 5

10.5 Perimeter and Area on the Coordinate Plane

Essential Question: How do you find the perimeter and area of polygons in the coordinate plane?

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Explain 1 Finding Areas of Figures on the Coordinate PlaneYou can use area formulas together with the Distance Formula to determine areas of figures such as triangles, rectangles and parallelograms.

triangle

A = 1 _ 2

bh

rectangleA = bh

parallelogramA = bh

rhombus

A = 1 _ 2

d 1 d 2

kite

A = 1 _ 2

d 1 d 2

trapezoid

A = 1 _ 2

( b 1 + b 2 ) h

Example 1 Find the area of each figure.

A Step 1 Find the coordinates of the vertices of △ABC.

A (-4, -2) , B (-2, 2) , C (5, 1)

Step 2 Choose a base for which you can easily find the height of the triangle.

Use _ AC as the base. A segment from the opposite vertex, B, to

point D (-1, -1) appears to be perpendicular to the base _ AC .

Use slopes to check.

slope of _ AC =

1 - (-2) _

5 - (-4) = 1 _ 3 ; slope of

_ BD = -1 - 2 _

-1 - (-2) = -3

The product of the slopes is 1 _ 3 ∙ (-3) = -1. _ BD is perpendicular

to _ AC , so

_ BD is the height for the base

_ AC .

Find the length of the base and the height.

AC = √ ―――――――――― (5 - (-4) ) 2 + (1 - (-2) ) 2 = √ ― 90 = 3 √ ― 10 ; BD = √ ―――――――――― (-1 - (-2) ) 2 + (-1 - 2) 2 = √ ― 10

Step 3 Determine the area of △ABC.

Area = 1 _ 2 bh = 1 _ 2 (AC) (BD) = 1 _ 2 ∙ (3 √ ― 10 ) ( √ ― 10 ) = 1 _ 2 ∙ 30 = 15 square units


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B Step 1 Find the coordinates of the vertices of DEFG.

D (-2, 6) , E (4, 3) , F (2, -1) , G (-4, 2)

Step 2 DEFG appears to be a rectangle. Use slopes to check that adjacent sides are perpendicular.

slope of _ DE : - _

4 - (-2) = _ 6 = ; slope of

_ EF : - 3 _

2 -

= _ =

slope of _ FG : 2 - _

-4 -

= _ = ; slope of _ DG : 2 - __

-

= _ =

so DEFG is a .

Step 3 Find the area of DEFG.

b = FG = √ ――――――――― (2 - ) 2 + ( - 2) 2 = √_

= √ ――

h = GD = ―――――――――――

( - (-4) ) 2

+ (6 - ) 2

= √_

= √ ――

Area of DEFG: A = bh = ( √ ―― ) ( √

―― ) = square units

Reflect

2. In Part A, is it possible to use another side of △ABC as the base? If so, what length represents the height of the triangle?

3. Discussion In Part B, why was it necessary to find the slopes of the sides?



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Your Turn

4. Find the area of quadrilateral JKLM with vertices J (−4, −2) , K (2, 1) , L (3, 4) , M (−3, 1) .

Explain 2 Finding Areas of Composite FiguresA composite figure is made up of simple shapes, such as triangles, rectangles, and parallelograms. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate. You can use the Area Addition Postulate to find the area of a composite figure.

Area Addition Postulate

The area of a region is equal to the sum of the areas of its nonoverlapping parts.

Example 2 Find the area of each figure.

A Possible solution: ABCDE can be divided up into a rectangle and two triangles, each with horizontal bases.

area of rectangle AGDE: A = bh = (DE) (AE) = (6) (4) = 24

area of △ABC: A = 1 _ 2 bh = 1 _ 2 (AC) (BF) = 1 _ 2 (8) (2) = 8

area of △CDG: A = 1 _ 2 bh = 1 _ 2 (CG) (DG) = 1 _ 2 (2) (4) = 4

area of ABCDE: A = 24 + 8 + 4 = 36 square units



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B PQRST can be divided into a parallelogram and a triangle.

△PQT appears to be a right triangle. Check that _ PT and are

perpendicular:

slope of _ PT : 1 - _

-3 -

= _ =

slope of : - 3 _

-3 -

= _ =

△PQT is a right triangle with base _ PT and height .

PT = √____

(-3 - ) 2

+ (1 - ) 2

= √_

= √____

(-3 - ) 2

+ ( -3) 2

= √_

= √_

area of △PQT: A = 1 _ 2 bh = 1 _ 2 ( √_

) ( √_

) =

_ QR || _ TS since both sides are vertical.

slope of = -1 _

-3 -

= _ = , so _ QT || . Therefore, QRST is a

parallelogram.

_ RT is an of △QRST and is horizontal. Because

_ RT ⊥ _ RQ , △QRT is a right

triangle with base and height . Therefore, the area of △QRST = 2 ∙ (area of △QRT) .

RT = , QR = , so the area of △QRT = 1 _ 2 ( ) ( ) = 6.

△QRST = 2 ∙ (area of QRT) = 2 ∙ = 12

area of PQRST: A = + = square units

Reflect

5. Discussion How could you use subtraction to find the area of a figure on the coordinate plane?



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Your Turn

6. Find the area of the polygon by addition. 7. Find the area of polygon by subtraction.

Explain 3 Using Perimeter and Area in Problem SolvingYou can use perimeter and area techniques to solve problems.

Example 3 Miguel is planning and costing an ornamental garden in the a shape of an irregular octagon. Each unit on the coordinate grid represents one yard. He wants to lay the whole garden with turf, which costs $3.25 per square yard, and surround it with a border of decorative stones, which cost $7.95 per yard. What is the total cost of the turf and stones?



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Analyze Information

Identify the important information.

• The vertices are A ( , 5) , B (1, ) , C (6, ) , D (4, ) , E (−1, −3) ,

F ( , −3) , G (−5, ) , and H ( , 2) .• The cost of turf is $ per square yard.

• The cost of the ornamental stones is $ per yard.

Formulate a Plan

• Divide the garden up into .

• Add up the of the smaller figures.

• Find the cost of turf by the total area by the cost per square yard.

• Find the perimeter of the garden by adding the of the sides.

• Find the cost of the border by the perimeter by the cost per yard.

• Find total cost by adding the and .

Solve

Divide the garden into smaller figures. The garden can be divided into square BCDE, kite ABEH, and parallelogram EFGH.

Find the area of each smaller figure.

area of BCDE:

slope of _ BC: -2

_ 6 -

=

slope of : -0 _

4 -

=

Also, BC = √____

( - 1) 2

+ (0 - ) 2

= √_

and

CD = √_____

(4 - ) 2

+ ( - ) 2

= √_

.

So BCDE is a square, with area A = s 2 = ( √_

yd) 2

= yd 2 .



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area of kite ABEH:

HA = √_____

(-1 - ( ) ) 2

+ ( - 2) 2

= √__

4 + = √_

AB = √ ――――――――――

( - (-1) ) 2

+ (2 - ) 2

= √__

+ 9 = √ ――

HE = √ ―――――――――――

(-1 - ( ) ) 2

+ ( - 2) 2

= √ ―――― + 25 = √

――

BE = √ ――――――――――

( -1) 2

+ (-3 - ) 2

= √ ―――― 4 + = √

――

So, ≅ and ≅ . Therefore ABEH is a kite.

b = d 1 = 8, h = d 2 = 4

A = 1 _ 2 d 1 d 2 = 1 _ 2 ( ) ( ) = yd 2

area of parallelogram EFGH:

and ̄ GH are both horizontal, so are parallel;

slope of _ EH :

2 - _

-3 - = _ = ; slope of :

2 - __

-

= _ =

So EFGH is a parallelogram, with base = and height. FH = .

area of EFGH: A = bh = ( ) ( ) = 2

Find the total area of the garden and the cost of turf.

area of garden: A = yd 2 + yd 2 + yd 2 = yd 2

cost of turf: ( yd 2 ) ( $ / yd 2 ) = $

Find the perimeter of the garden.

EF = 2 yd, GH = 2 yd

From area calculations, BC = CD = DE = √_

yd, and AB = AH =

FG = √_____

( - ) 2

+ ( - (-3) ) 2

= √_

,

perimeter of garden = GH + HA + AB + BC + CD + DE + EF + FG

= + + + + + + +

Find the cost of the stones for the border.

cost of stones: ( yd) ($ per yd) $

Find the total cost.

total cost: $ + $ = $



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yJustify and Evaluate

The area can be checked by subtraction:

area of large rectangle = (11) (10) = 110 square units

area = (11) ( ) - ( ) (3) - 1 _ 2 (2) ( ) - 1 _ 2 ( ) ( ) - (5) ( ) - 1 _ 2 ( ) (2) - 1 _ 2 ( ) (5)

- 1 _ 2 ( ) ( ) - ( ) ( ) - 1 _ 2 ( ) ( ) = _ - - - - - - - - =

The perimeter is approximately the perimeter of the polygon shown:

The perimeter of the polygon shown is ,

so the answer is reasonable.

Your Turn

8. A designer is making a medallion in the shape of the letter “L.” Each unit on the coordinate grid represents an eighth of an inch, and the medallion is to be cut from a 1-in. square of metal. How much metal is wasted to make each medallion? Write your answer as a decimal.



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Elaborate

9. Create a flowchart for the process of finding the area of the polygon ABCDEFG. Your flowchart should show when, and why, the Slope and Distance Formulas are used.

10. Discussion If two polygons have approximately the same area, do they have approximately the same perimeter? Draw a picture to justify your answer.

11. Essential Question Check-In What formulas might you need to solve problems involving the perimeter and area of triangles and quadrilaterals in the coordinate plane?



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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

Find the perimeter of the figure with the given vertices. Round to the nearest tenth.

1. D (0, 1) , E (5, 4) , and F (2, 6) 2. P (2, 5) , Q (-3, 0) , R (2, -5) , and S (6, 0)

3. M (-3, 4) , N (1, 4) , P (4, 2) , Q (4, -1) , and R (2, 2) 4. A (-5, 1) , B (0, 3) , C (5, 1) , D (4, -2) , E (0, -4) , and F (-2, -4)

Find the area of each figure.

5. 6.



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Find the area of each figure by addition.

7. 8.

Find the area of each figure by subtraction.

9. 10.



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11. Fencing costs $1.45 per yard, and each unit on the grid represents 50 yd. How much will it cost to fence the plot of land represented by the polygon ABCDEF?

12. A machine component has a geometric shaped plate, represented on the coordinate grid. Each unit on the grid represents 1 cm. Each plate is punched from an 8-cm square of alloy. The cost of the alloy is $0.43/c m 2 , but $0.28/c m 2 can be recovered on wasted scraps of alloy. What is the net cost of alloy for each component?

13. △ABC with vertices A(1, 1) and B(3, 5) has an area of 10 units 2 . What is the location of the third vertex? Select all that apply.

A. C(−5, 5)

B. C(3, −5)

C. C(−2, 5)

D. C(6, 1)

E. C(3, −3)


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14. Pentagon ABCDE shows the path of an obstacle course, where each unit of the coordinate plane represents 10 meters. Find the length of the course to the nearest meter.

Algebra Graph each set of lines to form a triangle. Find the area and perimeter.

15. y = 2, x = 5, and y = x 16. y = −5, x = 2, and y = −2x + 7

17. Prove that quadrilateral JKLM with vertices J(1, 5), K(4, 2), L(1, −4), and M(−2, 2) is a kite, and find its area.


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H.O.T. Focus on Higher Order Thinking

18. Explain the Error Wendell is trying to prove that ABCD is a rhombus and to find its area. Identify and correct his error. (Hint: A rhombus is a quadrilateral with four congruent sides.)

AB = √ ―――――――― (2 - (-2) ) 2 + (5 - 2) 2 = √ ― 25 = 5,

BC = √ ――――――― (6 - 2) 2 + (2 - 5) 2 = √ ― 25 = 5

CD = √ ―――――――― (2 - 6) 2 + (-1 - 2) 2 = √ ― 25 = 5,

AD = √ ―――――――――― (2 - (-2) ) 2 + (-1 - (2) ) 2 = √ ― 25 = 5

So _ AB ≅

_ BC ≅ _ CD ≅ _ AD , and therefore ABCD is a rhombus.

area of ABCD: b = AB = 5 and h = BC = 5, so A = bh = (5) (5) = 25

19. Communicate Mathematical Ideas Using the figure, prove that the area of a kite is half the product of its diagonals. (Do not make numerical calculations.)

20. Justify Reasoning Use the figure to derive the formula for the area of a trapezoid. Then use the Trapezoid Midsegment Theorem to show that the area of a trapezoid is the product of the length of its midsegment and its height.



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Lesson Performance Task

The coordinate plane shows the floor plan of two rooms in Fritz’s house. Because he enjoys paradoxes, Fritz has decided to entertain his friends with one by drawing lines on the floor of his tiled kitchen, on the left, and his tiled recreation room, on the right. The four sections in the kitchen are congruent to the four sections in the recreation room. Each square on the floor plan measures 1 yard on a side.

1. Find the area of each of the four sections of the kitchen. Add the four areas to find the total area of the kitchen.

2. Find the area of the kitchen by finding the product of the length and the width.

3. Find the area of the recreation room by finding the product of the length and the width.

4. Describe the paradox.

5. Explain the paradox.



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