introduction the distance formula can be used to find solutions to many real-world problems. in the...
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IntroductionThe distance formula can be used to find solutions to many real-world problems. In the previous lesson, the distance formula was used to find the distance between two given points. In this lesson, the distance formula will be applied to perimeter and area problems.
A polygon is a two-dimensional figure formed by three or more segments. Sometimes we need to calculate the perimeter or distance around a polygon, as well as find the area or the amount of space inside the boundary of a polygon. The distance formula is a valuable tool for both of these calculations.
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6.2.2: Calculating Perimeter and Area
Key Concepts• Situations where you would need to calculate perimeter
include finding the amount of linear feet needed to fence a yard or a garden, determining the amount of trim needed for a room, or finding the amount of concrete needed to edge a statue.
• Perimeter is the sum of the lengths of all the sides of a polygon.
• The final answer must include the appropriate label (units, feet, inches, meters, centimeters, etc.).
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6.2.2: Calculating Perimeter and Area
Key Concepts, continued• Sometimes the answer is not a whole number. If it is
not, you must simplify the radical and then approximate the value.
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6.2.2: Calculating Perimeter and Area
Calculating the Perimeter of a Polygon
1. Calculate the length of each side of the polygon using
the distance formula: .
2. Add all the lengths of the polygon to find the perimeter.
3. Simplify and approximate the value if necessary.
Be sure to include the appropriate label in your answer.
Key Concepts, continued• Calculating area is necessary when finding the
amount of carpeting needed for a room in your home, or to determine how large a garden will be.
• The area of a triangle is found using the formula
.
• The height of a triangle is the perpendicular distance from the third vertex to the base of the triangle.
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6.2.2: Calculating Perimeter and Area
Key Concepts, continued• It may be necessary to determine the equation of the
line that represents the height of the triangle before calculating the area. For an example of this, see Example 3 in the Guided Practice.
• Determining the lengths of the base and the height is necessary if these lengths are not stated in the problem.
• The final answer must include the appropriate label (units2, feet2, inches2, meters2, centimeters2, etc.).
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6.2.2: Calculating Perimeter and Area
Key Concepts, continued
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6.2.2: Calculating Perimeter and Area
Calculating the Area of a Triangle1. Find the equation of the line that represents the base of the triangle.
2. Find the equation of the line that represents the height of the triangle.
3. Find the point of intersection of the line representing the height of the triangle and the line representing the base of the triangle.
4. Calculate the length of the base of the triangle using the distance
formula: .
5. Calculate the length of the height of the triangle using the distance
formula: .
6. Calculate the area using the formula .
Be sure to include the appropriate label in your answer.
Key Concepts, continued• By definition, rectangles have adjacent sides that are
perpendicular.
• The area of a rectangle is found using the formula Area = (base)(height).
• The lengths of the base and height are found using the distance formula.
• The final answer must include the appropriate label (units2, feet2, inches2, meters2, centimeters2, etc.).
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6.2.2: Calculating Perimeter and Area
Calculating the Area of a Rectangle
1. Calculate the length of the base of the rectangle using the distance
formula: .
2. Calculate the length of the height of the rectangle using the distance
formula: .
3. Calculate the area using the formula .
Key Concepts, continued
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6.2.2: Calculating Perimeter and Area
))(( heightbaseA
Common Errors/Misconceptions• forgetting to simplify radicals
• incorrectly simplifying radicals
• adding x-values and y-values rather than subtracting them when using the distance formula
• incorrectly finding the height of a triangle
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6.2.2: Calculating Perimeter and Area
Example 1
Triangle ABC has vertices A (–3, 1), B (1, 3), and
C (2, –4). Calculate the perimeter of the triangle.
1. Calculate the length of each side of the triangle using the distance formula.
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6.2.2: Calculating Perimeter and Area
Length of AB=
Length of BC=
Length of AC=
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6.2.2: Calculating Perimeter and Area
22 )13()31(
22 )34()12(
22 )14()32(
22 )2()4( 416 20
22 )7()1( 491 50
22 )5()5( 2525 50
52
25
25
2. Calculate the perimeter of the triangle.
Perimeter
The perimeter of the triangle is about 18.6!
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6.2.2: Calculating Perimeter and Area
252552
21052
6.18
Guided Practice
Example 2Quadrilateral ABCD has vertices A (–3, 0), B (2, 4), C (3, 1), and D (–4, –3). Calculate the perimeter of the quadrilateral.
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
1. Calculate the length of each side of the quadrilateral using the distance formula.Calculate the length of .
Distance formula
Substitute (–3, 0) and (2, 4).
Simplify as needed.
The length of is units.
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continuedCalculate the length of .
Distance formula
Substitute (2, 4) and (3, 1).
Simplify as needed.
The length of is units. 15
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continuedCalculate the length of .
Distance formula
Substitute (3,1) and(–4,–3).
Simplify as needed.
The length of is units.
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continuedCalculate the length of .
Distance formula
Substitute (–4,–3) and(–3, 0).
Simplify as needed.
The length of is units.17
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
2. Calculate the perimeter of quadrilateral ABCD. Find the sum of the sides of the quadrilateral.
perimeter = AB + BC + CD + DA
The perimeter of quadrilateral ABCD is
units.18
6.2.2: Calculating Perimeter and Area
✔
Guided Practice
Example 3Triangle ABC has vertices A (-3, 2), B (4, 3), and C (5, –4). Calculate the area of triangle ABC.
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6.2.2: Calculating Perimeter and Area
A
B
C
Guided Practice: Example 3, continued
1. Find the equation of the line that represents the base of the triangle.Let be the base.
Calculate the slope of the equation that represents side .
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
Slope formula
Substitute (-3, 2) and (5, –4).
Simplify as needed.
The slope of the equation the represents side is .
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6.2.2: Calculating Perimeter and Area
)3(5
24
m
4
3
8
6
m
4
3
Guided Practice: Example 3, continuedWrite the equation of the line that represents side .
y – y1 = m(x – x1) Point-slope formula
y – y1 = (x – x1) Substitute for m.
y – 2 = (x – (-3)) Substitute (-3, 2) for (x1, y1)
y = x – Simplify.
The equation of the line that represents the base of the
triangle is .22
6.2.2: Calculating Perimeter and Area
4
34
3
4
3
4
34
1
4
1
4
3
xy
Guided Practice: Example 3, continued
2. Find the equation of the line that represents the height of the triangle.The equation of the line that represents the height is perpendicular to the base; therefore, the slope of this line is the opposite reciprocal of the base.
The slope of the line representing the height is .
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6.2.2: Calculating Perimeter and Area
3
4
Guided Practice: Example 3, continuedy – y1 = m(x – x1) Point-slope form
Substitute for m.
Substitute (4, 3) for (x1, y1).
Distribute over (x – 4).
Add 3 to both sides.
The equation of the line that represents the height of the triangle is
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6.2.2: Calculating Perimeter and Area
)(3
411 xxyy
3
4
)4(3
43 xy
3
43
16
3
43 xy
3
7
3
4 xy
3
7
3
4 xy
Guided Practice: Example 3, continued
3. Find the point of intersection of the line representing the height and the line representing the base of the triangle.Set the equation of the line representing the base and the equation of the line representing the height equal to each other to determine the point of intersection.
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
Set the equations equal to each o
Multiply by 12
Simplify
Add 9x
Add 28
Divide by 25
The point of intersection has an x-value of 1 .
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6.2.2: Calculating Perimeter and Area
4
1
4
3
3
7
3
4
xx
124
1
4
31212
3
7
3
412
xx
392816 xx
32825 x
2525 x1x
Guided Practice: Example 3, continued
Substitute 1 into either equation to find the y-value.
Equation of the line
representing base
Substitute 1 for x.
Simplify.
Solve for y. 27
6.2.2: Calculating Perimeter and Area
4
1
4
3
xy
4
1)1(
4
3
y
1y
4
1
4
3
y
Guided Practice: Example 3, continued
The point of intersection has a y-value of -1.
The point of intersection is (1,-1).
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6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
4. Calculate the length of the base, , of the triangle.
Distance formula
Substitute (-3,2) and (5, –4).
Simplify as needed.
The length of is 10 units.
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6.2.2: Calculating Perimeter and Area
22 )24()35(
22 )6()8(
101003664
Guided Practice: Example 3, continued
5. Calculate the length of the height from point B to the point of intersection.
Distance formula
Substitute (4, 3) and (1,-1) .
The length of the height is 5 units.
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6.2.2: Calculating Perimeter and Area
22 )41()31(
22 )3()4( 916 25 5
Guided Practice: Example 3, continued
6. Calculate the area of triangle ABC.
Area formula for triangles
Substitute the lengths of the height and the base of the triangle.
Simplify as needed.
The area of triangle ABC is 25 square units.
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6.2.2: Calculating Perimeter and Area
)5)(10(2
1A
25A