Lesson 6.1 Polygons
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Lesson 6.1Polygons
Today, we will learn to…> identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral
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# of Sides Name3
4
5
6
7
8
9
10
12
trianglequadrilateral
pentagonhexagonheptagon
octagonnonagon
decagon
dodecagon
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Theorem 6.1Interior Angles of a
Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is ______360°
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Section 6.1 Vocabulary
ConvexConcave
EquilateralEquiangular
RegularDiagonal
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Sides:
Vertices:
Diagonals:
S
T
U
DY
ST TU UD DY YS
S, T, U, D, Y
SU SD TD TY UY
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S
T
U
DY
There are 10 possible names of this pentagon.
STUDYSYDUTTUDYSTSYDUUDYSTUTSYD
DYSTUDUTSYYSTUDYDUTS
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How many diagonals can be drawn from N?
N M
O
PQ
R
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Starting with N, give 2 names for the hexagon.
N M
O
PQ
R
NMOPQR NRQPOM
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Is this a polygon? If not, explain. If so, is it
convex or concave?
Yes, it’s a convex
pentagon
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Is this a polygon? If not, explain. If so, is it
convex or concave?
No, polygons must be made of
segments
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Is this a polygon? If not, explain. If so, is it
convex or concave?
Yes, it’s a concave
dodecagon
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Is this a polygon? If not, explain. If so, is it
convex or concave?
No, polygons must be closed
figures
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Find x.
90 + 87 + 93 + x = 360x = 90
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Find x.
3x + 3x + 2x + 2x = 360x = 36
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Lesson 6.2Properties of Parallelograms
RULERS AND PROTRACTORS
Today, we will learn to…> use properties of parallelograms
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A quad is a parallelogram if and only if two pairs of opposite sides are parallel
parallelogram
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Draw a Parallelogram.
Measure each angle.Measure each side in centimeters.
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Theorems 6.2-6.5If a quadrilateral is a parallelogram, then…
1) 6.22) 6.33) 6.44) 6.5
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… opposite sides are __________congruent
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… opposite angles are__________.congruent
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… consecutive angles are__________.supplementary
1 2
34
m m m m
m m m m
1 2 180 1 4 180
3 2 180 3 4 180
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… diagonals __________each other.
bisect
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ABCD is a parallelogram. Find the missing angle and side measures.
1.A B
CD
105˚10
66
10
75˚
75˚
105˚
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ABCD is a parallelogram. Find AC and DB.
2. A
CD
8
85
B
5
AC = 10 DB = 16
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3. In ABCD, m C = 115˚. Find mA and mD.
4. Find x in JKLM.J K
LM(4x-9)˚
(3x+18)˚
mA = 115˚ mD = 65˚
x = 27
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ABCD is a parallelogram.
EC =
m BCD =
m ADC =
AD =
5
8
70° 110°
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The figure is a parallelogram.
x = y = 5 4
2x – 6 = 4 2y = 8
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The figure is a parallelogram.
x = y = 30 6 4x + 2x = 180 2y + 3 = y + 9
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The figure is a parallelogram.
x = y = 3 6
y
y
3x + 1 = 10 2y – 1 = y + 5
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The figure is a parallelogram.
x = y = 40 8 3x – 9 = 2x + 31 4y + 5 = 2y + 21
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Lesson 6.3Proving that Quadrilaterals
are Parallelograms
What is a converse?
Today, we will learn to…> prove that a quadrilateral is a
parallelogram
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Theorem 6.6
If both pairs of opposite sides are __________,
then it is a parallelogram.congruent
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Theorem 6.7If both pairs of opposite angles are __________,
then it is a parallelogram.congruent
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Is ABCD a parallelogram? Explain.
1. 2.A B
CD
10
6
10
6
A B
CDyes
no
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Theorem 6.8If an angle is
_______________ to both of its consecutive angles, then it is a parallelogram.
supplementary
1
2
3 m1 + m3 = 180˚m1 + m2 = 180˚
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Theorem 6.9If the diagonals
__________________, then it is a parallelogram.
bisect each other
AE = ECand
DE = EB
A
D
B
C
E
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Is ABCD a parallelogram? Explain.
3. 4. A B
CD
A B
CD
104˚
86˚ 104˚
no yes
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Theorem 6.10If one pair of opposite sides are ___________
and __________, then it is a parallelogram.
congruentparallel
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5.
8.
7.
6.
No Yes
Yes No
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9. List 3 ways to prove that a quadrilateral is a parallelogram
1) prove that both pairs of opposite sides are __________
2) prove that both pairs of opposite sides are __________3) prove that one pair of opposite sides are both ________ and ________
parallel
congruent
parallel congruent
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A ( , ) B ( , ) C ( , ) D ( , )
Prove that this is a parallelogram…
slope of AB isslope of BC isslope of CD isslope of AD is
0
4-2/5
-2/5
AB =BC =CD = AD =
4.15.44.15.4
2 3 4 -2 6 -3 2
4
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Lesson 6.4Special
Parallelograms
Today, we will learn to…> use properties of a rectangle,
a rhombus, and a square
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A square is a parallelogram with four congruent sides and four right angles.
A rhombus is a parallelogram with
four congruent sides.
A rectangle is a parallelogram with four right angles.four congruent sides. four right angles.
four congruent sides four right angles
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parallelograms
rhombuses rectangles
squares
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Sometimes, always, or never true?
1. A rectangle is a parallelogram.
2. A parallelogram is a rhombus.
3. A square is a rectangle.
4. A rectangle is a rhombus.
5. A rhombus is a square.
always true
sometimes true
always true
sometimes true
sometimes true
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Geometer’s Sketchpad
mAEB = 90CD = 4.48 cmBC = 4.48 cmAD = 4.48 cmAB = 4.48 cm
E
C
A B
DWhat do we know about the diagonals in a
rhombus?
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The diagonals of a rhombus are _____________.perpendicular
Theorem 6.11
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What do we know about the diagonals in a rhombus?
mECD = 40
mEDA = 50 mEDC = 50
mEAD = 40 mEAB = 40
mECB = 40mEBC = 50 mEBA = 50
E
C
A B
D
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The diagonals of a rhombus _____________________.bisect opposite angles
Theorem 6.12
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What do we know about the diagonals in a rectangle?
ED = 4.51 cmEB = 4.51 cm
EC = 4.51 cmEA = 4.51 cm
E
C
A B
D
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The diagonals of a rectangle are _____________. congruent
Theorem 6.13
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6. In the diagram, PQRS is a rhombus. What is the value of y?
2y + 3
5y – 6
P Q
RS
y = 3
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Find x. 7. rhombus
A
B
C
Dxº
52º
x = 38º
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Find m CDB. 8. rhombus
A
B
C
D32º
mCDB =32º
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Find AB.9. rectangle
A B
CD
10 12
AB = 16
?
202 = x2 + 122
10
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Find x.10. square
A B
CD
xº xº
x = 45˚
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Find EA & AB.11. square
EA =
A B
CD
4
EAB = 5.7
x2 = 42 + 42
x2 = 16 + 16x2 = 32x = 5.7
4
4
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Lesson 6.5Trapezoids
& Kites
Today, we will learn to…> use properties of trapezoids
and kites
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A trapezoid is a quadrilateral with only
one pair of parallel sides.
A B
D C
base
base
leg leg
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B A
D
C
Compare leg angles.
Geometer’s Sketchpad
mC = 65mD = 115mA = 90mB = 90
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In ALL trapezoids, leg angles are
_______________supplementary
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A trapezoid is an
isosceles trapezoid
if its legs are congruent.
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Geometer’s Sketchpad
Compare base angles.Compare leg angles.How do you know it is isosceles?
mA = 67 mD = 67 mC = 113 mB = 113 CD = 3.7 cmAB = 3.7 cm
A D
B C
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Theorem 6.14 & 6.15A trapezoid is isosceles if and
only if base angles are ___________.congruent
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Base angles are congruent.
A B
CDAC BD
The trapezoid is isosceles.
The triangles share CD.ADC BCD by SAS
CPCTC
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Theorem 6.16A trapezoid is isosceles if
and only if its diagonals are __________.congruent
AC BD
A B
CD
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ABCD is an isosceles trapezoid. Find the missing angle measures.
1. A B
CD100°
80° 80°
100°
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2. The vertices of ABCD are A(-1,2), B(-4,1), C(4,-3), and D(3,0). Show that ABCD is an isosceles trapezoid.
Figure is graphed on next slide.
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3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
D(3, 0)
C(4, -3)
B(-4, 1)
A(-1, 2)
AD || BC ?
AB =CD =
- ½ - ½
Legs are ? Diagonals are ? AC=BD =
50 10 10 50
OR?
Slope of AD isSlope of BC is
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x = 118 Find x.
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The midsegment is a segment that connects the midpoints of
the 2 legs of a trapezoid.
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Geometer’s SketchPad
EF = 8 cmCD = 12 cm
AB = 4 cm
EF = 7 cmCD = 11 cm
AB = 3 cm
A
EF = 5 cmCD = 6 cm
AB = 4 cm
EF = 7 cmCD = 9 cm
AB = 5 cm
FE
A B
D C
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Theorem 6.17Midsegment Theorem for
TrapezoidsThe midsegment of a
trapezoid is _________ to each base and its length is ______________ of the
bases.
parallel
the average
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Find x.
3. 4.
7
11
x
x
17
20
x = 9 x = 23
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KITE
A kite has two pairs of consecutive congruent
sides but opposite sides are not congruent and no sides
are parallel.
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Kite
What do we know if these points are equidistant from the endpoint of the segment?
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Theorem 6.18
In a kite, the longer
diagonal is the _________________
of the shorter diagonal.perpendicular bisector
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Kite
What do we know about congruent triangles?
How do we know the triangles are congruent?
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Kite
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Theorem 6.19In a kite, exactly one pair of opposite angles
are ________.congruent
The congruent angles are formed by the noncongruent sides.
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Find x and y.
5. 6.
5
x yx˚ 125˚
y˚
(y+30)˚29
x = 2 y = 2
x = 125
y = 40
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Theorem 6.19*
In a kite, the longer diagonal
________________.bisects opposite angles
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mJ =70°
mL = 70°
Find the missing angles.
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x =35
Find x.
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Find x.
x = 110
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Find x.
x = 5
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Based on our theorems, list all of the properties that must be true for the quadrilateral.
1. Parallelogram (definition plus 4 facts)
2. Rhombus (plus 3 facts)
3. Rectangle (plus 2 facts)
4. Square (plus 5 facts)
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Parallelogram
1) opposite sides are parallel
2) opposite sides are congruent
3) opposite angles are congruent
4) consecutive angles are supplementary
5) diagonals bisect each other
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Rhombus1) equilateral2) diagonals are perpendicular3) diagonals bisect opposite angles
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Rectangle1) equiangular2) diagonals are congruent
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Square1) equilateral2) equiangular3) diagonals are perpendicular4) diagonals bisect opposite angles5) diagonals are congruent
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Lesson 6.6Identifying Special
Quadrilaterals
Complete the chart of characteristics of special quadrilaterals.
Today, we will learn to…> identify special quadrilaterals
with limited information
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Given the following coordinates, identify the quadrilateral.
(-2, 1)(-2, 3)(3, 6) (0, 1)
kite
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Given the following coordinates, identify the quadrilateral.
(0, 0)(4, 0)(3, 7) (1, 7)
trapezoid
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Given the following coordinates, identify the quadrilateral.
rectangle
(-1, -3)(4, -3)(4, 3) (-1, 3)
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Given the following coordinates, identify the quadrilateral.
rhombus
(-2, 0)(3, 0)(6, 4) (1, 4)
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In quadrilateral WXYZ, WX = 15, YZ = 20, XY = 15,
ZW = 20. What is it?
It is a kite!
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Lesson 6.7Areas of Triangles and
Quadrilaterals
Today, we will learn to…> find the area of triangles and
quadrilaterals
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Postulate 22Area of a Square
Area = side2
A=s2
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Postulate 23Area Congruence Postulate
If two polygons are congruent, then they have the same area.
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Theorem 6.20Area of a Rectangle
Area = base ( height )
A = bh
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1. Find the area of the polygon made up of rectangles.
4 m
10 m
2 m
9 m
11 m
7 m11(2) = 22 m2
8(4) =
32 m2
5(4)= 20 m2
74 m2
?
??
Postulate 24
Area Addition Postulate
The area of a region is the sum of
the areas of its nonoverlapping
parts.
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Theorem 6.21Area of a Parallelogram
Area = base ( height)
A=bh
Do experiment.
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Theorem 6.22Area of a Triangle
A=½ bh
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Area of a Trapezoid
hh
b2
A = ½ h b1 + ½ h b2
b1
A = ½ h (b1 + b2)
A = ½ h b1 + ½ h b2
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Theorem 6.23Area of a Trapezoid
A = ½ height (sum of bases)
A=½ h (b1+b2)
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2. parallelogram 3. trapezoid
6
4 55 5
3
4
9
A = 6(4)
A = 24 units2A = ½ 4(9+3)
A = 24 units2
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Area of a Kite
b
b
x
y
A = ½ bx + ½ by
A = ½ b (x + y)What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2
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Theorem 6.24Area of a Kite
Area = ½ (diag.)(diag.)
A=½ d1 d2
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Area of a RhombusA = ½ bx + ½ by
A = ½ b(x + y)What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2
b
b
x
y
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Theorem 6.25Area of a Rhombus
Area = ½ (diag.)(diag.)
A=½ d1 d2
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4. Rhombus 5. Kite
4
35
34
A = ½ 6(8)
A = 24 units2
A = ½ 6(9)
A = 27 units2
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6. Rhombus 7. Trapezoid
8
x
A = 80 units2
x = 5
A = 55 units2
h = 5
h
13
9
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8. Find the total area.
15
8 A = ½(10)(8+20)
A = 440 units2
20
25A = 140
A = 20(15)
A = 300
?10
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A = 12(11)
blue A = ½ (12)(5)
11
12
A = 132
132 = 122 + x2
x = 513
just blue?
blue A = 30
pink A = 132 – 60
pink A = 72
2 blue regions A = 60
?5
9. Find the areas of the blue and pink regions.
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