Aim #3 - Manhasset Schools

2-12-15
Aim 3: What is the Scaling Principle for Area?
Do Now: Find the area of each figure.
Figure (2)
Figure (1)
Figure (4)
Figure (3)
Exploratory Challenge: Complete the table below.
Dimensions of Original Figure
Area of Dimensions of Area of Scale Factor
Original Figure
Similar Figure Similar Figure
Fig.(1) b = 8, h = 3
3
Fig.(2) b = 5, h = 3
2
Ratio of Area of Similar Figure to Original Figure
Fig.(3) b = 5, h = 4
Fig.(4) b = 3, h = 2
Make a conjecture about the relationship between the areas of the original
figure and the similar figure with respect to the scale factor between the
figures.
The Scaling Principle for Triangles:
If similar triangles S and T are related by a scale factor of r, then the
respective areas are related by a factor of r2.
Proof of the Scaling Principle for Triangles
Triangle S
Triangle T
Area of S =______ Area of T = ______
Ratio of
=
=
HW #2:
1) 52 + 52 + .5(6)(4) = 62 units squared.
2) Area larger semi circle = 32π
Area large triangle = 64√3
Area unshaded semi circle = (32/9)π
Area unshaded small triangle = (64√3)/9
A = (32π - (32/9)π) + (64√3 - (64√3)/9) ≈ 188 units squared
3) 62 + 62 - (3)2 = 63 units squared.
4) tan 35 = 110/x
x ≈ 157 ft
The Scaling Principle for Polygons:
If similar polygons P and Q are related by a scale factor of r, then the
respective areas are related by a factor of r2.
Polygon Q
Polygon P
Polygon Q is the image of Polygon P under a similarity transformation with scale
factor r. How can we compute the area of a polygon like this?
Proof of the Scaling Principle for Polygons
Any polygon can be subdivided into non-overlapping triangles.
Polygon Q
Polygon P
Each of the side lengths in polygon Q is r times the corresponding lengths in
polygon P. Polygon P is divided into triangles T1, T2,T3, T4,T5
Polygon Q
Polygon P
By the Scaling Principle for Triangles, the areas of each of the triangles in Q is
r2 times the areas of the corresponding triangles in P.
Area (Q) = r2T1 + ______ + _______ + _______ + _________
= r2(
Area (Q) = r2(
+
+
)
+
+
)
The Scaling Principle for Area:
If similar figures A and B are related by a scale factor of r, then their
respective areas are related by a factor of r2.
Exercise (1): Rectangles A and B are smilar and are drawn to scale. If the area
of rectangle A is 88 mm2, what is the area of rectangle B?
Horizontal and Vertical Scaling
• By what scale factor was the unit square scaled horizontally? _____ How
does the area of the resulting rectangle compare to the area of the unit
square?
• What is happening between the second and third image?
• How are the two directions of the scaling related to each other?_________
How does the area of the third image relate to the two scale factors that
were applied?
Summary: When a figure is scaled by factors a and b in two perpendicular
directions, then its area is multiplied by a factor of ab.
• The sides of a 1 x 1 square are scaled horizontally and vertically by factors a
and b, respectively. The new area is ____
1
1
Area =1
• The triangle with base 1 and height 1 is scaled by factors a and b,
respectively. The area of the scaled triangle is ______
1
1
Area =
• What would be the effect on the original figure if a scale factor is between 0
and 1?
Exercises:
1. If the scale factor between two similar figures is 1.2, what is the scale factor
of their respective areas?
2. If the scale factor between two similar figures is
, what is the scale factor
of their respective areas?
3. In the following figure, AE and BD are segments.
a) Why is ΔABC ~ ΔEDC?
b) What is the scale factor of the similarity transformation
that takes ΔABC to ΔEDC?
c) What is the value of the ratio of the area of ΔABC to the area of ΔEDC?
Justify your answer.
d) If the area of ΔABC is 30 cm2, what is the area of ΔEDC? (nearest cm2)
4. In ΔABC, DE connects two sides of the triangle and is parallel to BC. If the
area of ΔABC is 54 and BC = 3DE, find the area of ΔADE.
5. The small star has an area of 5. The large star is obtained from the small star
by stretching by a factor of 2 in the horizontal direction and by a factor of 3 in
the vertical direction. Find the area of the large star.
5 x 2 x 3 = 30 in2
6. Figure T' is the image of figure T that has been scaled horizontally by a scale
factor of 4, and vertically by a scale factor of 1/3, If the area of T' is 24 units2,
what is the area of figure T?
AT x 4 x 1/3 = 24
AT = 18 units squared
Let's Sum it Up!
• The Scaling Principle for Triangles: If similar triangles and are related by a
scale factor of r, then the respective areas are related by a factor of r2 .
• The Scaling Principle for Polygons: If similar polygons and are related by a
scale factor of r, then their respective areas are related by a factor of r2.
• The Scaling Principle for Area: If similar figures and are related by a scale
factor of r, then their respective areas are related by a factor of r2.
Name ______________________
CC Geometry H
Date ______________________
Hwk. #3
1. Figures E and F are similar and are drawn to scale. If the area of Figure E is
120 mm2, what is the area of figure F?
2. A a rectangle has an area of 18. Fill in the table below by answering the
questions that follow. The first row has been completed for you.
1
2
Original Original
Dimensions Area
18 x 1
18
9x2
18
6x3
18
½ x 36
18
3
Scaled
Dimensions
9x½
4
Scaled
Area
5
Scaled Area
6
Area ratio in terms
Original Area of the scale factor
¼
18
a. Enter in Column 3: If the given rectangle is dilated from a vertex with a scale
factor of ½, what are the dimensions of the images of each of your rectangles?
b. Enter in Column 4: What are the areas of the images of your rectangles?
c. Enter in Column 5: How do the areas of the images of your rectangles compare
to the area of the original rectangle? Write each ratio in simplest form.
d. Enter in Column 6: Write the values of the ratios of areas entered in column 5
in terms of the scale factor ½.
e. If the areas of two unique rectangles are the same, x, and both figures are
dilated by the same scale factor r, what can we conclude about the areas of the
dilated images?
OVER
3. Find the ratio of the areas of each pair of similar figures. The lengths of
corresponding line segments are shown.
b.
a.
c.
4. An isosceles trapezoid has base lengths of 12 in. and 18 in. If the area of the
larger shaded triangle is 72 in2, find the area of the smaller shaded triangle.
12 inches
18 inches
5. In ΔABO, A'B' ll AB, OA' = 3, A'A = 6, OB' = 4, and B'B = 8. Find the ratio of
the area of ΔOA'B' to the area of quadrilateral ABB'A'.
O
3
A'
6
4
B'
8
A
B
6. What is the effect on the area of a rectangle if...
a. its height is doubled and base left unchanged?
b. its base and height are both doubled?
c. its base were doubled and height cut in half?