Download 2205 Unit 3 part C NOTES

Geometry 2205
Unit 3:
Mrs. Bondi
Unit 3: Two-Dimensional Shapes
Lesson Topics:
Lesson 1: Congruent Figures (PH text 4.1)
Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2)
Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3)
Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4)
Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5)
Lesson 6: Congruence in Right Triangles (PH text 4.6)
Lesson 7: Congruence in Overlapping Triangles (PH text 4.7)
Lesson 8: Midsegments of Triangles (PH text 5.1)
Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2)
Lesson 10: Bisectors in Triangles (PH text 5.3)
Lesson 11: Medians and Altitudes (PH text 5.4)
Lesson 12: Indirect Proof (PH text 5.5)
Lesson 13: Inequalities in One Triangle (PH text 5.6)
Lesson 14: Inequalities in Two Triangles (PH text 5.7)
Lesson 15: The Polygon-Angle Sum Theorems (PH text 6.1)
Lesson 16: Properties of Parallelograms (PH text 6.2)
Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3)
Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4)
Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5)
Lesson 20: Trapezoids and Kites (PH text 6.6)
Lesson 21: Polygons in the Coordinate Plane (PH text 6.7)
Lesson 22: Applying Coordinate Geometry (PH text 6.8)
Lesson 23: Proofs Using Coordinate Geometry (PH 6.9)
Lesson 24: Proportions in Triangles (PH text 7.5)
Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1)
Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2)
Lesson 27: Areas of Regular Polygons (PH text 10.3)
Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4)
Lesson 29: Trigonometry and Area (PH text 10.5)
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Geometry 2205
Unit 3:
Mrs. Bondi
Get ready for ch.7, p.429, and ch.10, p.611
Additional Review:
1. The lengths of two sides of a polygon are in the ratio 2 : 3. Write expressions for the measures of the two sides in terms
of the variable x.
2. ∆HJK ~ ∆RST. Complete each statement.
3. To the nearest inch, a door is 75 in. tall and 35 in. wide. What is the ratio of the width to the height?
4. What is a proportion that has means 9 and 10 and extremes 6 and 15?
5. Find the geometric mean of each pair of numbers. a. 4 and 25
b. 9 and 12
Solve for the value of the variables in each right triangle.
6.
7.
8.
9.
10.
11.
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Geometry 2205
Unit 3:
Mrs. Bondi
Lesson 24: Proportions in Triangles (PH text 7.5)
Objective:
to use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem
Reminder:
Theorem 7-4
Side-Splitter Theorem
If a line is parallel to one side of a triangle
and intersects the other two sides, then it
divides those sides proportionally.
Example 1: Find the value of x.
Example 2: Find the value of x.
Reminder:
Corollary to Side-Splitter Theorem If three parallel lines
intersect two transversals, then the
segments intercepted on the transversals
are proportional.
Example 3: Find the value of x and y.
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Geometry 2205
Unit 3:
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Theorem 7-5
Triangle-Angle-Bisector Theorem
If a ray bisects an angle of a triangle,
then it divides the opposite side into
two segments that are proportional to
the other two sides of the triangle.
Examples:
4. Find the value of x.
9
5. Find the value of x.
5
6
15
X
3
6. Find the value of x.
7. Find the value of x.
3 cm
15 cm
7 cm
8
x
30
Practice:
HW: p.474 #8-30 even, 31-35 (skip 24)
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Geometry 2205
Unit 3:
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Geometry 2205
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Geometry 2205
Unit 3:
Mrs. Bondi
Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1)
Objective:
to find areas of parallelograms and triangles
Theorem 10-1
Area of a Rectangle
A = _________
Theorem 10-2
Area of a Parallelogram
A = _________
Theorem 10-3
Area of a Triangle
A = _________
Base of a parallelogram
Altitude
Height
Examples:
1. Find the area and perimeter of:
15 cm
2. Find the area of the shaded triangular region.
12 cm
3 ft
20 cm
2 ft
2 ft
3. Abby’s Bakery has a plan for a 50’ by 31’ rectangular parking lot. The four parking spaces are
congruent parallelograms (10 ft by 16 ft), the driving area is a rectangle, and the two unpaved areas for
flowers are congruent triangles.
a) Complete the diagram based on the description.
50'
31'
b) Explain two different ways to find the area of the region that must be paved.
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Geometry 2205
Unit 3:
Mrs. Bondi
4. In a triangle, a base and the corresponding height are in the ratio 3:2. If the area is 108 in2,
determine the length of the base and its corresponding height.
D
5. Determine the value of x in parallelogram ABCD.
C
X
20"
18"
A
Practice:
HW: p.619 #8-20 even, 41-43
8
30"
B
Geometry 2205
Unit 3:
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Geometry 2205
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Geometry 2205
Unit 3:
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Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2)
Objective:
to find the area of a trapezoid, rhombus or kite
Trapezoid -
base
b1
base –
leg
leg
h
leg –
b2
base
height –
Theorem 10-4
Area of a Trapezoid
The area of trapezoid is half the product of the height and the sum of the lengths
of the bases.
b1
b2
1
A  hb1  b2 
2
h
b2
b1
Examples:
1.
Find the area of a trapezoid with a height of 42 in. and bases measuring 121 in. and 145 in.
2.
B
C
Find the area of Trapezoid ABCD.
BC = 21 cm
AD = 25 cm
60
A
D
Isosceles Trapezoid -
>
P
T
11
Z
>
A
E
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Geometry 2205
Unit 3:
Mrs. Bondi
>
P
Examples:
3. Find the area of trapezoid TRAP
if ET = 15 ft and AP = 10 ft.
>
T
H
A
45
10 2 ft
45
E
6 cm
R
T
4. Find the area of isosceles trapezoid NRTH.
Leave your answer in simplest radical form.
60
N
2 cm
O
5. The area of a trapezoid is 160 cm 2 . Its height is
8 cm and the length of its shorter base is 14 cm.
Find the length of the longer base.
Theorem 10-5
Area of a Kite
The area of rhombus or a kite is half the product of the lengths of its diagonals.
A
Problem:
a)
b)
1
d1d 2
2
Find the measures of the numbered angles in the rhombus. (justify each)
1
A  d1 d 2
Find the area. BE = 5cm; CE = 6 cm
B
C
2
1
Justify each step.
E
4
3
2
A
D
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Geometry 2205
Unit 3:
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K
Examples:
6. Find the area of kite KITE if EI = 10 ft.,
KM = 3 ft., and KT = 14 ft.
E
M
T
7.
Find the area of the given figure.
Practice:
HW: p.626 #12-34 even, 37-39 (skip 28)
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Geometry 2205
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Lesson 27: Areas of Regular Polygons (PH text 10.3)
Objective:
to find the area of a regular polygon
Regular Polygon –
Center –
Radius –
Apothem –
Postulate 10-1
If two figures are congruent, then their areas are equal.
Theorem 10-6
Area of a Regular Polygon
The area of a regular polygon is half the product of the apothem and the
perimeter.
A=
1
ap
a=
A  ap or A 
2
2
p=
Examples:
1)
Find the area of a regular heptagon with a 12 inch apothem, and 10 inch sides.
2)
Find the area of a regular decagon with a 9.4 inch apothem, and 9 inch sides.
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Geometry 2205
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Mrs. Bondi
1)
The length of an apothem of an equilateral triangle is 7 cm. Find the area.
2)
The length of a radius of a regular hexagon is 8 cm. Find the area.
Practice:
HW: p.632 #8-28 even, 44-45
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Geometry 2205
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Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4)
Objective:
to find the relationships between the similarity ratio and the perimeters and
areas of similar figures.
Theorem 10-6
If the scale factor of two similar figures is a:b, then:
a)
the ratio of their perimeters is a:b
b)
the ratio of their areas is a2:b2
Example 1
Find the area of a regular hexagon with side length 3 cm, then find the area of a
regular hexagon with side lengths of 6 cm.
3cm
3cm
6cm
Example 2
The areas of two similar rectangles are 48 in2 and 75 in2. What is the ratio of
their perimeter (scale factor)?
Example 3
Madeline used 144 tiles to tile a rectangular kitchen floor. If each dimension of
the kitchen were doubled, what is the ratio of the perimeters?
X
What is the ratio of the areas?
Y
How many tiles would Madeline need to cover the bigger floor?
2X
2Y
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Geometry 2205
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Mrs. Bondi
Example 4
The areas of two similar rectangles are 1875 ft2 and 135 ft2.
Find the ratio of their perimeters.
Example 5
The similarity ratio of the dimensions of two similar pieces of window
glass is 3:5. If the smaller piece cost $2.50, what should be the cost of the larger
piece (based on area)?
Practice:
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Practice:
HW: p.638 #10-30 even, 31, 42-44
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Geometry 2205
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Lesson 29a: Trigonometry and Area (PH text 10.5)
Objective:
to find areas of regular polygons and triangles using trigonometry
The apothem of any regular polygon forms isosceles triangles. You can use trigonometry to help find
the area of any regular polygon as long as you know the length of a side, radius or apothem.
Reminder:
Area of a Regular Polygon
A
1
ap
2
or
A
ap
2
Finding area using trigonometry:
Steps:
1. Draw in one apothem and the associated isosceles triangle.
2. Imagine all of these triangles formed within the figure.
3. Divide to find the measure of the angle formed in the right angle
at the center, x.
4. Use trigonometry, the given information and the angle measure
to find the needed information to compute the area.
5. Compute the area of the polygon. It is best to write out what you
will plug into the calculator so you can double check your
accuracy.
Examples:
1
2.
3.
HW. p.646 #6-12 even, 20, 24-26
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Geometry 2205
Unit 3:
Mrs. Bondi
2. hexagon with side length 9m
3. decagon with radius 4mm
4. octagon with radius 10cm
5. 20-gon with radius 5 in.
6. 15-gon with perimeter 90 ft
7. 18-gon with radius 126 m
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Geometry 2205
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Lesson 29b: Trigonometry and Area (PH text 10.5)
Objective:
to find areas of regular polygons and triangles using trigonometry
Examples/Practice:
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Geometry 2205
Unit 3:
Mrs. Bondi
Additional Practice: Geometric Probability
The polygons below are all regular polygons. Find the area of the shaded region.
Then find the probability of a pebble, being randomly tossed, hitting the white area.
HW: p.646 #
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Geometry 2205
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Mrs. Bondi
Review 3.24-29 (text 7.5 & 10.1-5)
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13. What is the area of the 12-gon with a radius of 8 cm?
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