Download - Geometry Section 11-1/11-2
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EXTENDING AREA, SURFACE AREA, AND VOLUME
CHAPTER 11/12
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AREAS OF PARALLELOGRAMS, TRIANGLES, RHOMBI, AND
TRAPEZOIDS
SECTION 11-1 AND 11-2
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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?
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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:
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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
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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
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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
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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
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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
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EXAMPLE 1
Find the perimeter and area of .!RSTU
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EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
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EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
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EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
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EXAMPLE 1
Find the perimeter and area of .!RSTU
P = 2l + 2w
P = 2(32)+ 2(20)
P = 64 + 40
P = 104 in.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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?
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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
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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
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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
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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
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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
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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.
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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.
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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)
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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
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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
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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
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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.
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POSTULATE 11.2
If two figures are congruent, then they have the same area.
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EXAMPLE 3
Find the area of the trapezoid.
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EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
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EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
A = 12(1)(3+ 2.5)
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EXAMPLE 3
Find the area of the trapezoid.
A = 12h(b
1+b
2)
A = 12(1)(3+ 2.5)
A = 12(5.5)
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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
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EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
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EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
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EXAMPLE 4
Fuzzy Jeff designed a deck shaped like the trapezoid shown. Find the area of the deck.
a2 +b2 = c2
42 +b2 = 52
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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
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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
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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
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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
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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
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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
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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)
![Page 55: Geometry Section 11-1/11-2](https://reader030.vdocuments.us/reader030/viewer/2022013013/58ab0cf21a28ab70038b5189/html5/thumbnails/55.jpg)
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)
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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)
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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
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EXAMPLE 5
Find the area of each rhombus or kite.
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(7)(12)
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(7)(12)
A = 42 ft2
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EXAMPLE 5
Find the area of each rhombus or kite.
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(14)(18)
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EXAMPLE 5
Find the area of each rhombus or kite.
A = 12d
1d
2
A = 12(14)(18)
A = 126 in2
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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?
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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
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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
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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
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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 )
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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
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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
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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
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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
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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
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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.
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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.
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PROBLEM SET
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PROBLEM SET
p. 767 #1, 2, 5-9 all; p. 777 #1-13 odd
“The best preparation for tomorrow is doing your best today.” - H. Jackson Brown, Jr.