Warm-up
Use Geometer’s Sketchpad to construct 2 non-overlapping,
similar triangles whose corresponding side lengths are in
the ratio of 3 to 2, and display the perimeter of each triangle.
In questions 1 – 4, identify whether the triangles can be proven similar,
and state the similarity theorem that supports the conclusion.
B
2.
L
1.
X
3
N
Z
6
4.5
M
K
O
A
Yes, AA~
4
R
10
3.
4.
15
T
P
12
8
Yes, SAS~
Y
C
U
W
M
18
V
Q
Yes, SSS~
N
Yes, AA~
5. In the diagram below, rectangle RSTU is dilated (center P) and its
image is rectangle RSTU . What is the scale factor of this dilation?
1:3
6. A blueprint of a triangular garden with sides measuring 2, 6, and 7 inches
is placed on an overhead projector and projected onto a screen, as shown
below. The shortest side of the triangle on the screen is 10 inches.
How many inches is the longest
side of the triangle on the screen?
35 inches
C
7. In the diagram, triangle ABC is isosceles. Point P is chosen on
base AB so that BP is twice AP. Segments PM and PN are
drawn perpendicular to legs AC and BC , respectively.
M
B
a. Prove that APM and BPN are similar.
b. If AM = 4, what is the length of BN ?
N
P
A
8
c. Assuming the ratio of BP to AP remains 2:1, prove that CN = CM – AM.
CN = CB – BN = CA – BN = CM + AM – BN = CM + AM – 2AM = CM – AM
C
the diagram,
is isosceles.
Point
P is chosen on
1. Atriangle
 B ABC
(Isosceles
Triangle
Theorem)
PM and
PMA
PNM
congruent
rightPN
angles.
se AB so2.that
BP isand
twice
AP. are
Segments
are
3. AMP toBNP
(AA
) BC , respectively.
wn perpendicular
legs AC
and
Prove that APM and BPN are similar.
If AM = 4, what is the length of BN ?
N
M
B
A
P
8. In the last homework assignment, you conjectured that in the regular
pentagon shown, the ratio of AB to AD is equal to the ratio of AD to AP.
Prove this conjecture.
(5  2)180
 108
5
Since the pentagon is regular, AD  BD making
ABD isosceles, so BAD = ABD = 36
In the regular pentagon, ADB = CAD =
Prove:
AB AD

AD AP
Similarly, ACD isosceles , so ACD = ADC = 36
Therefore, ADB  APD (AA~)
A
A
A
36
36
36
C
36
P
36
D
P
36
B
AB AD

AD AP
36
D
D
(Definition of
similar triangles)
36
B
Which of the following triangles
are similar to ABC (there may
be more than one answer).
A
G
C
B
D
E
F
 a. AFE
b. DBE
c. DFC
d. AED
e. GFC
f. ABD
Which of the following triangles
are similar to ABC (there may
be more than one answer).
A
G
C
B
D
E
F
 a. AFE
 b. DBE
c. DFC
d. AED
e. GFC
f. ABD
Which of the following triangles
are similar to ABC (there may
be more than one answer).
A
G
C
B
D
E
F
 a. AFE
 c. DFC
 b. DBE
e. GFC
f. ABD
d. AED
Which of the following triangles
are similar to ABC (there may
be more than one answer).
 a. AFE
 c. DFC
 b. DBE
e. GFC
f. ABD
A
A
G
20
d. AED
C
G
B
70
B
D
E
F
D
55
C
35
E
F
Which of the following triangles
are similar to ABC (there may
be more than one answer).
 a. AFE
 c. DFC
 e. GFC
A
G
C
B
D
E
F
 b. DBE
d. AED
f. ABD
Which of the following triangles
are similar to ABC (there may
be more than one answer).
 a. AFE
 c. DFC
 e. GFC
A
G
C
B
D
E
F
 b. DBE
d. AED
f. ABD
10. Recall that the median of a trapezoid connects the midpoints of the non-parallel
sides and its length is the average of the lengths of the bases. Is it possible for
the diagonals of an isosceles trapezoid to be the same length as the median of
the trapezoid? Explain.
It is not possible.
AB  DC
, then AB + DC = 2(MN) =2(AC).
2
Since the diagonals of an isosceles trapezoid are congruent, AC + BD = 2(AC).
Suppose AC = MN
Since we know MN 
Therefore, AB + DC = AC + BD
So AB + DC = (AP + PC) + (BP + PD) = AP + BP + PC + PD
> DC
> AB
Contradiction
C
D
P
M
A
N
B
by the triangle inequality
(the sum of the lengths
of any two sides of a
triangle is greater than
the length of the third
side.)
1
1
1
AB, CR = AC, and BQ = BC.
3
3
3
Use Geometer’s Sketchpad to construct the diagram.
11. In the diagram, AP =
AREA
The ratio of the perimeters of 2 similar
triangles is equal to the scale factor.
The ratio of the areas of 2 similar triangles
is equal to the square of the scale factor.
Area of a rectangle
A = bh
h
W
bL
Area of a rectangle
A = bh
h
b
b
Area of a parallelogram, rhombus,
rectangle, and square: A = bh
h
b
1
Area of a triangle: A = bh
2
h
b
L
h1
h
bL2
A = ½ L 1L2
b
Not What it Seems
An Area Paradox
A = ½(13)(5)
A = 32.5
5
13
What is the area of the right triangle shown?
A = ½(13)(5)
A = 32.5
A = ½(13)(5)
A = 32.5
A = ½(13)(5)
A = 32.5
A = ½(13)(5)
A = 32.5
A = ½(13)(5)
A = 32.5
A = 32.5–1 = 31.5
What’s the area now?
What is the area of the square shown?
A = 64
A = 64
A = 64
A = 64
Area = 65
What’s the area now?
How is this possible?
A = 64
b2
1 (b + b )
Area of a trapezoid: A = 2 h 1
2
h
b1
b2
1 (b + b )
Area of a trapezoid: A = 2 h 1
2
h
median
A = (median)(altitude)
b1
Practice:
16 inches
1. A square has the same area as the parallelogram
shown. What is the perimeter of the square?
9 inches
48 inches
C
2. CM is a median of ABC. Explain why the areas
of AMC and BMC must be equal.
Theorem: The median of a triangle divides
the triangle in to triangles of equal area.
A
M
B
3. The lengths of the bases of a certain trapezoid are 13 and 10. If the area
of the trapezoid is 92, what is the length of the altitude of the trapezoid?
8 inches