Transcript 6-1 Using Proportions • I. Ratios and Proportions Ratio- comparison of two or more quantities Example: 3 cats to 5 dogs 3:5 3 to 5 3/5 Proportion: two equal.

6-1 Using Proportions
• I. Ratios and
Proportions
Ratio- comparison of
two or more quantities
Example: 3 cats to 5 dogs
3:5
3 to 5
3/5
Proportion: two equal ratios:
Example: 3/5 = 6/10
Equality of
Cross Products
A = C if and only if AD=BC
B D
A and D are the extremes
B and C are the means
II. Examples
• 1. X
4
= 20
5
2. X+ 3 = 4
9
3
Solutions:
x • 5 = 20 • 4
3(x + 3) = 4•9
5x = 80
3x + 9 = 36
x = 16
3x = 27
x=9
• 3. A + 1
A- 1
= 5
6
6(A + 1) = 5(A – 1)
6A + 6 = 5A – 5
A + 6 = -5
A = -11
• 4. Find 32 % of
156.
_is_ = _%_
of
100
_x_
156
= _32_
100
100x = 4992
x = 49.92
• 5. Write a proportion to find x.
20
X +2
X + 2 = _4_
20
16
4
16( X + 2) = 4(20)
16X + 32 = 80
16X = 48
X=3
16
a:b:c represents a:b b:c a:c
6. The ratio of the measures of three sides of a triangle is 8:7:5 and the
perimeter of the triangle is 240 cm. Find the measure of each side
of the triangle
8(12) = 96
8x + 7x + 5x = 240
7(12) = 84
20x = 240
5(12) = 60
X = 12
Complete P. 341: 6, 7, 9, 11 – 13, 15
6. 1:75 or 1/75
7. 170:9 or 170/9
9. X = 14
11. Yes
12. Yes
13. No
15. 3x + 4x + 5x = 72
12x = 72
x=6
3(6) = 18
4(6) = 24
5(6) = 30
6-2 Exploring Similar
Polygons
• I. Similar polygons
– All corresponding sides must be
proportional and all corresponding
angles must be congruent
– Symbol is ~
II. Scale factors and
dilations
• Scale factor: ratio of two
corresponding lengths
• Dilation: a transformation that
reduces or enlarges
ENLARGEMENT
REDUCTION
Ratio of the sides (scale factor) a:b
Ratio of the perimeters
(same as scale factor)
a:b
Ratio of the Areas
(remember: area is
always squared)
a²:b²
III. Examples
• 1. Find the scale factor for the
20
similar polygons
8
6
6
8
15
15
20
The scale factor of the smaller to the larger is 6/15 or 8/20 (both = 2/5)
The scale factor of the larger to the smaller is 15/6 or 20/8 (both = 5/2)
2. Polygon RSTUV and polygon ABCDE
are similar.
E
3
V
A
x
D
5
B4 C
U
R
y+2
S
18
T
a. Find the scale factor of
RSTUV to ABCDE
18/4 = 9/2
b. Find the value of x
c. Find the value of y
18/4 = x/3
18/4 = (y+2)/5
18(3) = 4x
18(5) = 4(y+2)
13.5 = x
90 = 4y + 8
82 = 4y y = 20.5
• 3. Perform a dilation of 3 on the
square with coordinates:
• D(0,0) E (5,0) F(0,5) G (5,5)
(0,5)
(0,0)
(5,5)
(0,15)
(15,15)
(5,0)
(0,0)
(15, 0)
Complete p.349 and p.350: 1, 2, 4, 8 - 11
1. A. Yes, all corresponding angles are congruent and all
corresponding sides are proportional (1:1)
B. No, sides are proportional not necessarily congruent
2. They both could be right: Larger to smaller or smaller to larger
4. The corresponding angles must be congruent
8. Sometimes. The sides might not be proportional and the
acute angles might not be congruent
9. Always: if figures are congruent they are similar
10. A. 6/9 = 2/3
11. (0,0) (9,0) (0,12)
B. 6/9 = 12/x
6/9 = 14/y
yes, scale factor of
x = 18 in.
Y = 21 in.
old to new is 1/3
C. 63 in.
D. 2/3
6-3 Identifying Similar Triangles
• I. Methods that show if triangles are similar
• AA Similarity
• SSS Similarity
• SAS Similarity
II. AA Similarity
• If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar.
III. SSS Similarity
• If the measures of the corresponding
sides of two triangles are proportional,
then the triangles are similar.
IV. SAS Similarity
• If the measures of two sides of a triangle
are proportional to the measures of two
corresponding sides of another triangle
and the included angles are congruent,
then the triangles are similar.
V. Examples
A
2
• 1. Find CB
AB // DE
D
Since the lines are //, the two
sets of corresponding angles
are congruent and the triangles
are similar by AA.
6
C
3
E
6/8 = 3/x
8
6
6x = 8(3)
3
x
x = 4 so CB = 4
B
2. Find the value of x, AE, and ED
4
A
B
AB // CD
3x + 4
E
Since the lines are parallel,
The two pair of alternate interior
angles are congruent and the
triangles are similar by
AA.
C
X + 12
8
D
So, 4/8 = (3x + 4)/(x + 12)
Then 4(x + 12) = 8(3x + 4)
4x + 48 = 24x + 32 x = 4/5 and AE = 3(4/5) + 4 = 6.4
ED = 4/5 + 12 = 12.8
Complete p. 357: 2, 4, 6-11
2. Only one; you know the right angles are congruent and if you know
one pair of acute angles are congruent then the third angles are
automatically congruent.
4. They are both correct. In both cases the cross product is the same
6. Yes, they are similar by AA
7. Not enough information
8. True, SSS similarity
9. Triangle AEC and Triangle BDC by AA
10. AA similarity 3/12 = x/(x+12) x = 4
3/12 = y/20 y = 5
11. SAS similarity 2/4 = x/9 x = 4.5
6-4 Parallel Lines and
Proportional Parts
• I. Triangle proportions
• If a line is parallel to one side of a
triangle and intersects the other two sides
in two distinct points, then it separates
these sides into segments of proportional
lengths.
This line makes a smaller similar
triangle to the outer big triangle
• The converse is also true- if you have two
similar triangles, a line intersecting is
parallel.
II. Midpoints
• A segment whose endpoints are the
midpoints of two sides of a triangle is
parallel to the third side of the triangle,
and its length is one-half the length of the
third side.
III. Many transversals
• If three or more parallel lines intersect
two transversals, then they cut off the
transversals proportionally.
IV. Examples
1. You want to plant a row of trees along a slope as shown.
Find the indicated lengths.
Assume all of the vertical lines are parallel.
XY = 88
X
a
b
a/10 = 88/66 a = 13 1/3
c
b/12 = 88/66 b = 16
d
c/10 = 88/66 c = 13 1/3
e
10
12
10
16
18
d/16 = 88/66 d = 21 1/3
e/18 = 88/66 e = 24
2.
Complete the following statements:
a. a/b = c/___
b. c/e = d/___
c. a/e = b/___
b
a
d
c
f
Answers:
a. d
b. f
c. f
e
P
3. Complete the following:
a. PQ/PR = QS/___
Q
b. PQ/QR = PS/___
R
c. PR/PT = PQ/___
S
Answers
a. RT
b. ST
c. PS
T
4. Find the value of x and y
y+ 2
3y -8
3x - 9
x+2
Answers:
2(3x – 9) = x + 2
6x – 18 = x + 2
y + 2 = 3y – 8
10 = 2y
5=y
5x = 20
x=4
Complete page 366: 6 - 12
6. A. LT
B. RL
7. A. True B. False, RS = 16
8. x = 2 and y = 12
9. Yes
10. No, must have DG // EF
11. Yes
12. x = 3 1/3 feet y = 2 2/3 feet
z = 2 feet
6-5 Parts of Similar Triangles
• I. Perimeters
• If two triangles are similar, then the perimeters are
proportional to the measures of corresponding sides.
II. Altitudes
• Corresponding altitudes are proportional
to the measures of the corresponding
sides.
III. Angle bisectors
• Corresponding angle bisectors are
proportional to the measures of the
corresponding sides.
IV. Medians
• Corresponding medians are proportional
to the measures of the corresponding
sides.
V. Angle bisector
An angle bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two sides.
Examples:
1. Triangle ABC is similar to triangle DEF. Find the perimeter of
triangle ABC.
D
A
9
41
40
B
C
E
9
F
Answer:
AB/DE = Perimeter ABC/ Perimeter DEF
9/40 = x/90
x = 20.25
2. Find EH. Triangle ABC is similar to Triangle DEF.
B
E
5
4
A
G
30
C
x
D
Answer: 4/x = 5/30
x = 24
H
F
3. Complete the proportion for the similar triangles:
a. a/d = c/___
a
b
b. b/c = e/___
c
c. a/b = d/___
e
d
f
Answers:
a. f
b. f
c. e
4. Find the value of x and y for the similar triangles.
2
x+2
5
3x - 4
3y - 4
2y + 12
Answers: 2/5 = (x + 2) / (3x – 4)
2/5 = (3y – 4) / (2y + 12)
2(3x – 4) = 5(x + 2)
x = 18
2(2y + 12) = 5(3y- 4) y = 4
5. Find the value of x.
3
12
X
14
12/3 = 14/x
x = 3.5
Complete page 373: 5 - 9
5. AB: angle bisector theorem
6. DF: The medians of two similar triangles
are proportional to two corresponding sides.
7. X = 15
x = 2 isn’t
possible because
8. X = 6.75
9. x/5 = 2/(7 – x)
x(7 – x) = 5(2)
the leg and
7x - x² = 10
0 = x² - 7x + 10
hypotenuse of a
0 = (x – 5) (x – 2)
right triangle
x = 5 or 2
are not equal.