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Δ A B C is similar to Δ X Y Z . The lengths of two sides of each triangle are given in the figure.

The above image shows two similar triangles. The smaller triangle is labeled A B C. The length of two sides is given for the smaller triangle A B C. The length from A to B is 17. The length from B to C is a. The length from C to D is 15. The larger triangle is labeled X Y Z. The length is given for two sides. The length from X to Y is 25.5. The length from Y to Z is 12. The length from Z to X is y.

Find the length of side a .

8

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Δ A B C is similar to Δ X Y Z . The lengths of two sides of each triangle are given in the figure.

Find the length of side y .

22.5

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The next example shows how similar triangles are used with maps.

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

The above image shows two similar triangles and how they are used with maps. The smaller triangle on the left shows San Francisco, Las Vegas and Los Angeles on the three points. San Francisco to Los Angeles is 1.3 inches. Los Angeles to Las Vegas is 1 inch. Las Vegas to San Francisco is 2.1 inches. The second larger triangle shows the same points. The distance from San Francisco to Los Angeles is x. The distance from Los Angeles to Las Vegas is 270 miles. The distance from Las Vegas to San Francisco is not noted.

Solution

Read the problem. Draw the figures and label with the given information. The figures are shown above.
Identify what we are looking for. The actual distance from Los Angeles to San Francisco.
Name the variables. Let x = distance from Los Angeles to San Francisco.
Translate into an equation. Since the triangles
are similar, the corresponding sides are
proportional. We'll make the numerators
"miles" and the denominators "inches."
.
Solve the equation. .
.
Check.
On the map, the distance from Los Angeles to
San Francisco is more than the distance from
Los Angeles to Las Vegas. Since 351 is more
than 270 the answer makes sense.
.
Answer the question. The distance from Los Angeles to San Francisco is 351 miles.

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On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

The above image is a triangle with one side labeled “Seattle, 4.5 inches”. The other side is labeled “Portland 3.5 inches”. The third side is labeled 1.5 inches. The vertex is labeled “Boise.”

150 miles

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Using the map above, find the distance from Portland to Boise.

350 miles

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We can use similar figures    to find heights that we cannot directly measure.

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Solution

Read the problem and draw a figure. .
We are looking for h , the height of the tree.
We will use similar triangles to write an equation.
The small triangle is similar to the large triangle. .
Solve the proportion. .
Simplify. .
Check.
Tyler's height is less than his shadow's length so it makes
sense that the tree's height is less than the length of its shadow.
.

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A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

40 feet

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A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

60 feet

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Key concepts

  • Property of Similar Triangles
    • If Δ A B C is similar to Δ X Y Z , then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
  • Problem Solving Strategy for Geometry Applications
    1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
    2. Identify what we are looking for.
    3. Name what we are looking for by choosing a variable to represent it.
    4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
    5. Solve the equation using good algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

Questions & Answers

A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Kendra Reply
Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
What is observation
adeyemi Reply
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Sunnyshay Reply
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Sophia Reply
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs $9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?
Parker Reply
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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