geometry section 11-1/11-2
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EXTENDING AREA, SURFACE AREA, AND VOLUME
CHAPTER 11/12
AREAS OF PARALLELOGRAMS, TRIANGLES, RHOMBI, AND
TRAPEZOIDS
SECTION 11-1 AND 11-2
ESSENTIAL QUESTIONS• How do you find perimeters and areas of
parallelograms?
• How do you find perimeters and areas of triangles?
• How do you find areas of trapezoids?
• How do you find areas of rhombi and kites?
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
Can be any side of a parallelogram
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
Can be any side of a parallelogram
The perpendicular distance between any two parallel bases of a parallelogram
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
Can be any side of a parallelogram
The perpendicular distance between any two parallel bases of a parallelogram
Can be any side of a triangle
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
Can be any side of a parallelogram
The perpendicular distance between any two parallel bases of a parallelogram
Can be any side of a triangle
The length of a segment perpendicular to a base to the opposite vertex
VOCABULARY
1. Base of a Parallelogram:
2. Height of a Parallelogram:
3. Base of a Triangle:
4. Height of a Triangle:
5. Height of a Trapezoid:
Can be any side of a parallelogram
The perpendicular distance between any two parallel bases of a parallelogram
Can be any side of a triangle
The length of a segment perpendicular to a base to the opposite vertex
The perpendicular distance between bases
EXAMPLE 1
Find the perimeter and area of .!RSTU
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
a2 = 256
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
a2 = 256
a = 16
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
a2 = 256
a = 16h = 16
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
a2 = 256
a = 16h = 16
A = 32(16)
EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
A = bh a2 +b2 = c2
a2 +122 = 202
a2 +144 = 400−144 −144
a2 = 256
a2 = 256
a = 16h = 16
A = 32(16)
A = 512 in2
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + c
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83 Matt needs 12 boards.
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
A = 12(12)(9)
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
A = 12(12)(9)
A = 54 ft2
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
A = 12(12)(9)
A = 54 ft2
549
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
A = 12(12)(9)
A = 54 ft2
549
= 6
EXAMPLE 2
Matt Mitarnowski needs to buy enough boards to make the frame of the triangular sandbox shown and enough sand to cover the bottom. If one of the boards is 3 feet long and one bag of sand
covers 9 square feet of the sandbox, how many boards and bags does he need to buy?
P = a +b + cP = 12+16+ 7.5
P = 35.5 ft
35.53
≈11.83
A = 12bh
Matt needs 12 boards.
A = 12(12)(9)
A = 54 ft2
549
= 6
Matt needs 6 bags of sand.
POSTULATE 11.2
If two figures are congruent, then they have the same area.
EXAMPLE 3
Find the area of the trapezoid.
EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
A = 12(1)(3+ 2.5)
EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
A = 12(1)(3+ 2.5)
A = 12(5.5)
EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
A = 12(1)(3+ 2.5)
A = 12(5.5)
A = 2.75 cm2
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
b1= 9; b
2= 9− 3 = 6
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
b1= 9; b
2= 9− 3 = 6
A = 12h(b
1+b
2)
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
b1= 9; b
2= 9− 3 = 6
A = 12h(b
1+b
2)
A = 12(4)(6+ 9)
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
b1= 9; b
2= 9− 3 = 6
A = 12h(b
1+b
2)
A = 12(4)(6+ 9)
A = (2)(15)
EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
16+b2 = 25−16 −16
b2 = 9
b2 = 9 b = 3
b1= 9; b
2= 9− 3 = 6
A = 12h(b
1+b
2)
A = 12(4)(6+ 9)
A = (2)(15)
A = 30 ft2
EXAMPLE 5
Find the area of each rhombus or kite.
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(7)(12)
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(7)(12)
A = 42 ft2
EXAMPLE 5
Find the area of each rhombus or kite.
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(14)(18)
EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(14)(18)
A = 126 in2
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
256 = x2
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
256 = x2
256 = x2
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
256 = x2
256 = x2
x = 16
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
256 = x2
256 = x2
x = 16
d1= 16 in.
EXAMPLE 6
One diagonal of a rhombus is half as long as the other diagonal. If the area of the rhombus is 64 square inches, what are the lengths
of the diagonals?
A = 12d
1d
2
d1= x
d2= 1
2x
64 = 12(x )( 1
2x )
64 = 14x2
4(64) = ( 14x2 )4
256 = x2
256 = x2
x = 16
d1= 16 in.
d2= 8 in.
PROBLEM SET
PROBLEM SET
p. 767 #1, 2, 5-9 all; p. 777 #1-13 odd
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