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Lesson 6-4 Parallel Lines and Proportional Parts
Lesson 6-5 Parts of Similar Triangles
Lesson 6-6 Fractals and Self-Similarity
Example 2 Extended Ratios in Triangles
Example 3 Solve Proportions by Using Cross Products
Example 4 Solve Problems Using Proportions
The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth.
To find this ratio, divide the number of athletes by the total number of students.
0.3 can be written as
Answer: The athlete-to-student ratio is 0.3.
The country with the longest school year is China with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use
365 as the number of days in a year.)
Answer: 0.7
Multiple- Choice Test Item
In a triangle, the ratio of the measures of three sides is 5:12:13, and the perimeter is 90 centimeters. Find the measure of the shortest side of the triangle.
A 15 cm B 18 cm C 36 cm D 39 cm
Read the Test Item
You are asked to apply the ratio to the three sides of the triangle and the perimeter to find the shortest side.
Solve the Test Item
We can rewrite 5:12:13 as 5 x :12 x :13 x and use those measures for the sides of the triangle. Write an equation to represent the perimeter of the triangle as the sum of the measures of its sides.
Perimeter
Combine like terms.
Divide each side by 30.
Use this value of x to find the measures of the sides of the triangle.
The shortest side is 15 centimeters. The answer is A.
Check Add the lengths of the sides to make sure that the perimeter is 90.
Answer: A
Multiple- Choice Test Item
In a triangle, the ratio of the measures of three sides is 3:4:5, and the perimeter is 42 feet. Find the measure of the longest side of the triangle.
A 10.5 ft B 14 ft C 17.5 ft D 37 ft
Answer: C
Solve
Answer: 27.3
Original proportion
Cross products
Multiply.
Divide each side by 6.
Solve
Answer: –2
Original proportion
Cross products
Simplify.
Add 30 to each side.
Divide each side by 24.
Solve each proportion.
a.
Answer: 4.5
b.
Answer: 9
A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model.
Because the scale model of the boxcar and the boxcar are in proportion, you can write a proportion to show the relationship between their measures. Since both ratios compare feet to inches, you need not convert all the lengths to the same unit of measure.
Substitution
Cross products
Multiply.
Divide each side by 40.
Answer: The width of the model is 3.6 inches.
Two large cylindrical containers are in proportion.
The height of the larger container is 25 meters with a diameter of 8 meters. The height of the smaller container is 7 meters. Find the diameter of the smaller container.
Answer: 2.24 m
Example 3 Proportional Parts and Scale Factor
Example 4 Enlargement of a Figure
Example 5 Scale Factors on Maps
Determine whether the pair of figures is similar.
Justify your answer.
Q
The vertex angles are marked as 40 º and 50º, so they are not congruent.
Since both triangles are isosceles, the base angles in each triangle are congruent. In the first triangle, the base angles measure and in the second triangle, the base angles measure
Answer: None of the corresponding angles are congruent, so the triangles are not similar.
Determine whether the pair of figures is similar.
Justify your answer.
T
Thus, all the corresponding angles are congruent.
Now determine whether corresponding sides are proportional.
The ratios of the measures of the corresponding sides are equal.
Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so
Determine whether the pair of figures is similar.
Justify your answer.
a.
Answer: Both triangles are isosceles with base angles measuring 76 º and vertex angles measuring 28º.
The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent,
Determine whether the pair of figures is similar.
Justify your answer.
b.
Answer: Only one pair of angles are congruent, so the triangles are not similar.
An architect prepared a 12-inch model of a skyscraper to look like a real 1100-foot building.
What is the scale factor of the model compared to the real building?
Before finding the scale factor you must make sure that both measurements use the same unit of measure.
1100(12) 13,200 inches
Answer: The ratio comparing the two heights is
The scale factor is , which means that the model is the height of the real skyscraper.
A space shuttle is about 122 feet in length. The
Science Club plans to make a model of the space shuttle with a length of 24 inches. What is the scale factor of the model compared to the real space shuttle?
Answer:
The two polygons are similar. Write a similarity
statement. Then find x, y, and UV.
Use the congruent angles to write the corresponding vertices in order.
Now write proportions to find x and y.
To find x :
Similarity proportion
Cross products
Multiply.
Divide each side by 4.
To find y :
Similarity proportion
Cross products
Multiply.
Subtract 6 from each side.
Divide each side by 6 and simplify.
Answer:
The two polygons are similar. Find the scale factor
of polygon ABCDE to polygon RSTUV.
The scale factor is the ratio of the lengths of any two corresponding sides.
Answer:
The two polygons are similar.
a. Write a similarity statement. Then find a, b, and ZO.
Answer: ; b.
Find the scale factor of polygon TRAP to polygon .
Answer:
Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5.
If the length and width
of rectangle PQRS are 10 meters and 4 meters, respectively, what are the length and width of
rectangle WXYZ?
Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be
Answer:
Quadrilateral GCDE is similar to quadrilateral JKLM
with a scale factor of If two of the sides of GCDE measure 7 inches and 14 inches, what are the
lengths of the corresponding sides of JKLM?
Answer: 5 in., 10 in.
The scale on the map of a city is inch equals 2 miles. On the map, the width of the city at its widest point is inches. The city hosts a bicycle race across town at its widest point. Tashawna bikes at
10 miles per hour. How long will it take her to complete the race?
Explore Every equals 2 miles. The distance across the city at its widest point is
Plan Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time.
Solve
Cross products
Divide each side by 0.25.
The distance across the city is 30 miles.
Divide each side by 10.
It would take Tashawna 3 hours to bike across town.
Examine To determine whether the answer is reasonable, reexamine the scale. If 0.25 inches 2 miles, then 4 inches 32 miles. The distance across the city is approximately 32 miles. At 10 miles per hour, the ride would take about 3 hours. The answer is reasonable.
Answer: 3 hours
An historic train ride is planned between two landmarks on the Lewis and Clark Trail. The scale on a map that includes the two landmarks is 3 centimeters = 125 miles. The distance between the two landmarks on the map is 1.5
centimeters. If the train travels at an average rate of 50 miles per hour, how long will the trip between the landmarks take?
Answer: 1.25 hours
Example 1 Determine Whether Triangles Are Similar
Example 2 Parts of Similar Triangles
In the figure, and Determine which triangles in the figure are similar.
Angles Theorem.
Vertical angles are congruent, by the Alternate Interior
Answer: Therefore, by the AA Similarity Theorem,
In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5
.
Determine which triangles in the figure are similar.
I
Answer:
ALGEBRA Given
QT 2 x 10, UT 10, find RQ and QT.
Since because they are alternate interior angles. By AA Similarity,
Using the definition of similar polygons,
Substitution
Cross products
Now find RQ and QT .
Distributive Property
Subtract 8 x and 30 from each side.
Divide each side by 2.
Answer:
ALGEBRA Given
and CE x + 2, find AC and CE.
Answer:
INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a
12-foot light pole and measured its shadow at 1
P.M.
The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower?
Assuming that the sun’s rays form similar triangles, the following proportion can be written.
Now substitute the known values and let x be the height of the Sears Tower.
Substitution
Cross products
Simplify.
Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.
INDIRECT MEASUREMENT
On her trip along the East coast,
Jennie stops to look at the tallest lighthouse in the U.S. located at
Cape Hatteras, North Carolina.
At that particular time of day,
Jennie measures her shadow to be 1 feet 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?
Answer: 196 ft
Example 1 Find the Length of a Side
Example 2 Determine Parallel Lines
Example 3 Midsegment of a Triangle
Example 4 Proportional Segments
In and Find SU.
S
From the Triangle Proportionality Theorem,
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
Answer:
In and Find BY.
B
Answer: 15.75
In and Determine whether Explain.
In order to show that we must show that
Since the sides have proportional length.
Answer: lengths, since the segments have proportional
In and AZ = 32.
Determine whether Explain.
X
Answer: No; the segments are not in proportion since
Triangle ABC has vertices A ( –2, 2), B (2, 4,) and C (4, –4).
is a midsegment of Find the coordinates of
D and E.
(2, 4)
(-2, 2)
(4, –4)
Use the Midpoint Formula to find the midpoints of
Answer: D (0, 3), E (1, –1)
Triangle ABC has vertices A ( –2, 2), B (2, 4) and C (4, –4).
is a midsegment of Verify that
(2, 4)
(-2, 2)
(4, –4)
If the slopes of slope of slope of
Answer: Because the slopes of
Triangle ABC has vertices A ( –2, 2), B (2, 4) and C (4, –4).
is a midsegment of Verify that
(2, 4)
(-2, 2)
(4, –4)
First, use the Distance Formula to find BC and DE.
Answer:
Triangle UXY has vertices U ( –3, 1), X (3, 3), and Y (5, –7).
is a midsegment of
a. Find the coordinates of W and Z .
Answer: W (0, 2), Z (1, –3) b. Verify that
Answer: Since the slope of and the slope of c. Verify that
Answer: Therefore,
In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in city blocks
that the streets are apart. Find x.
Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.
Triangle Proportionality Theorem
Cross products
Multiply.
Divide each side by 13.
Answer: 32
In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in city blocks
that the streets are apart. Find x.
Answer: 5
Find x and y.
To find x :
Given
Subtract 2 x from each side.
Add 4 to each side.
To find y :
The segments with lengths are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal.
Answer: x = 6; y = 3
Equal lengths
Multiply each side by 3 to eliminate the denominator.
Subtract 8 y from each side.
Divide each side by 7.
Find a and b.
Answer: a = 11; b = 1.5
Example 1 Perimeters of Similar Triangles
Example 3 Medians of Similar Triangles
Example 4 Solve Problems with Similar Triangles
If and find the perimeter of
C
Let x represent the perimeter of The perimeter of
Answer: The perimeter of
Proportional
Perimeter Theorem
Substitution
Cross products
Multiply.
Divide each side by 16.
If and
RX = 20 , find the perimeter of
R
Answer:
and Find the ratio of the length of an altitude of to the length of an altitude of are similar with a ratio of According to Theorem 6.8, if two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides.
Answer: The ratio of the lengths of the altitudes is
and Find the ratio of the length of a median of to the length of a median of
Answer:
In the figure, is an altitude of
and is an altitude of Find x if and
K
Write a proportion.
Cross products
Divide each side by 36.
Answer: Thus, JI = 28.
In the figure, is an altitude of
and is an altitude of Find x if and
N
Answer: 17.5
The drawing below illustrates two poles supported by wires. , , and Find the height of the pole .
are medians of since and
If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. This leads to the proportion measures 40 ft. Also, since both measure 20 ft.
Therefore,
Write a proportion.
Cross products
Simplify.
Divide each side by 80.
Answer: The height of the pole is 15 feet.
The drawing below illustrates the legs, of a
table. The top of the legs are fastened so that AC measures 12 inches while the bottom of the legs open
such that GE measures 36 inches. If BD measures 7
inches, what is the height h of the table?
Answer: 28 in.
Example 3 Evaluate a Recursive Formula
Example 4 Find a Recursive Formula
Example 5 Solve a Problem Using Iteration
below is found by connecting the midpoints of the sides of Prove that
Given: E, D, and F are midpoints of respectively.
Prove:
C
Proof:
Statements
1.
E, D, and F are midpoints of respectively.
2.
Reasons
1.
Given
2.
Triangle Midsegment
Theorem
3.
3.
Alternate Interior Angles
Theorem
5.
6.
Proof:
Statements
4.
Reasons
4.
Corresponding Angles
Postulate
5.
Transitive Property
6.
AA Similarity
is formed by connecting the midpoints of the sides and of Prove that
Given: Z and X are the midpoints of respectively.
Prove: w
4.
5.
Proof:
Statements
1.
Z and X are the midpoints of respectively.
Reasons
1.
Given
2.
2.
Triangle Midsegment
Theorem
3.
3.
Corresponding Angles
Postulate
4.
Reflexive Property
5.
AA Similarity
Draw an equilateral triangle. Create a fractal by drawing another equilateral triangle within it and shading above or beneath the triangle that shares the horizontal side of the equilateral triangle. Stages 1 and 2 are shown.
Answer:
Draw a square. Create a fractal by bisecting the top and left side of the square and drawing a smaller square inside the larger square as shown.
Answer:
Find the value of where x initially equals 1 .
Then use that value as the next x in the expression.
Repeat the process three more times and describe your observations.
The iterative process is to square the value of x , multiply that value by 3, and then subtract 1. Begin with The value of becomes the next value of x .
x 1
2
2
11
11
362
362
393,131
Answer: 2, 11, 362, 393,131; x values increase with each iteration, approaching infinity.
Find the value of where x initially equals 0 .
Then use that value as the next x in the expression.
Repeat the process three more times and describe your observations.
Answer: 3, –15, –447, –399,615; x values decrease with each iteration, approaching negative infinity.
The diagram below represents the odd integers 1, 3, 5, and 7 . Find a formula in terms of the row number for the sum of the values in the diagram.
Notice that each sum is the row number squared.
Answer:
What is the sum of the first 10 odd numbers?
The sum of the values in the tenth row will be 10 2 or 100.
Answer: 100
Examine the pattern shown below.
Row Sum
1
2
3
4
a. Find a formula for the sum of the values in the diagram.
Answer: S n
= S n
–1 n b.
Find the number in row 7.
Answer: 28
BANKING Joaquin has $1500 in a savings account that earns 4.1% interest. If the interest is compounded annually, find the balance of his account after 4 years.
First, write an equation to find the balance after one year.
Answer: After 4 years, Joaquin will have $1761.55 in his account.
BANKING Anna has $3500 in a savings account that earns 3.8% interest. If the interest is compounded annually, find the balance of her account after 5 years.
Answer: $4217.50
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