4. Similar Triangles
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Chapter 11
4.
Similar Triangles
| Student Learning Objectives: |
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Why
Similar triangles have the same shape but may differ in size. The Angle-Angle Similarity Theorem states that if two triangles have two pairs of angles that are equal, the triangles are similar. This concept is used in geometric problems to solve for unknown sides or angles through indirect measurement. Slope, the measure of steepness of a line, plays a key role in identifying similarity in triangles involving lines. Understanding the similarity of slope triangles allows for comparing the properties of two triangles and solving problems related to their size and angles. These concepts are useful in fields like engineering, architecture, and design, where accurate measurements and scaling are important. Knowing how to apply similarity transformations and the Angle-Angle Similarity Theorem can help solve real-world geometric problems.
Lesson Settings & Tools
| | 12 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Similar Triangles
Slide of 12
Triangle properties are applied in a wide range of real-life situations. This lesson emphasizes similarity and proportionality principles in triangles. Applying these concepts are very useful when direct measurements are impossible to record.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Geometric Transformations and Similarity:
Geometric Transformations and Similarity:
Triangles and Measurement:
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Explore
The Slope Triangle of a Line
Explore the slope triangle on the displayed line by dragging the point above the line to adjust the dimensions of the slope triangle.
- What is the ratio of the rise to the run?
- Observe how this ratio changes (if at all) when modifying the dimensions of the slope triangle.
- Based on these observations, what conclusions can be drawn about the relationship between the dimensions of the slope triangle and the ratio of the rise to the run?
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Discussion
Angle-Angle Similarity Theorem
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Pop Quiz
Similar Triangles?
Determine if the given triangles are similar using the Angle-Angle Similarity Theorem. If necessary, apply the Interior Angles Theorem to find missing angles before identifying corresponding angles.
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Discussion
Indirect Measurement
The method of Indirect measurement uses the properties of similar triangles and proportionality to calculate distances or the dimensions of objects that are difficult to measure directly. For instance, visualize two flag poles under the sunlight; one 15 feet tall pole casts a 12-foot shadow and the other casts an 8-foot shadow. What is the height of the second flagpole?
In what is called shadow reckoning, the angles made by the sun's rays off of different objects to a certain spot are equal. Similar triangles are formed by the flagpoles and their shadows as there are two pairs of congruent angles between them.
h/15=8/12
MultEqn
LHS * 15=RHS* 15
h=8/12*15
MoveRightFacToNum
a/c* b = a* b/c
h=8*15/12
Multiply
Multiply
h=120/12
CalcQuot
Calculate quotient
h=10
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Example
The Shadow of a Tree
Dylan is a skater who enjoys exploring the math behind the ramps, obstacles, and shadows at a local skate park. One sunny day, he took a break under a tall tree and noticed his shadow next to the tree's shadow. This sparked his curiosity about the tree's height, and he decided to use its shadow to calculate it.
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Hint
The ratio of the lengths of corresponding sides of similar triangles is always the same.
Solution
Begin by labeling the vertices for each of the triangles in the diagram.
In this diagram, AD corresponds to AE, and BD corresponds to CE. As the triangles ABD and ACE are similar, the ratio between their corresponding sides will be the same. AD/AE=BD/CE Note that AE equals the sum of AD and DE. AD/AD+DE=BD/CE In this case, AD, BD, AD, and DE are known. Substitute these values into the proportion and solve for CE to find the height of the tree.
AD/AD+DE=BD/CE
SubstituteValues
Substitute values
13/13+ 66=6.5/CE
AddTerms
Add terms
13/79=6.5/CE
MultEqn
LHS * CE=RHS* CE
13/79* CE=6.5
MultEqn
LHS * 79=RHS* 79
13 * CE=513.5
DivEqn
.LHS /13.=.RHS /13.
CE=39.5
Dylan determined that the height of the tree is 39.5 feet.
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Example
Dylan's Ramp Adventure
Dylan is enjoying his time skating at the park. He drops-in on a triangular-shaped ramp. He begins his descent from a height of 15 feet, which is also the height of the ramp. After he glides down the slope for 20 feet, he realizes that he is now only 70 % as high as when he started.
He wants to know the total length of the slope, which means the total distance he will glide down. Help him find out this information.
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Hint
Solution
The problem states that the height of a ramp is 15 feet. Dylan rides down 20 feet from the top of the ramp and lands at a height that is 70 % of his initial height. Dylan's current height can be found by multiplying his initial height by 0.70.
Dylan's Current Height 0.70* 15=10.5ft
This means that his current height is 10.5 feet. Use the base of the ramp, its height, the distance that Dylan covered, Dylan's current height, and the slope of the ramp to draw a diagram of the situation.
Based on the diagram, it can be seen that both triangles ABC and DBE have a right angle each and share a common angle at vertex B. Therefore, these triangles are similar because of the Angle-Angle Similarity Theorem. △ ABC~ △ DBE Consequently, DE corresponds to AC and BE corresponds to BC. DE&↔& AC BE&↔& BC Additionally, BC equals the sum of CE and BE. Since the ratios between the lengths of corresponding sides of the triangles are equal, the following equation holds true. DE/AC=BE/BC ⇓ DE/AC=BE/CE+BE In this equation, only BE is unknown. The known values can be substituting in this proportion to find the value of BE.
DE/AC=BE/CE+BE
SubstituteValues
Substitute values
10.5/15=BE/20+BE
CalcQuot
Calculate quotient
0.7=BE/20+BE
MultEqn
LHS * (20+BE)=RHS* (20+BE)
0.7(20+BE)=BE
Distr
Distribute 0.7
14+0.7 * BE=BE
SubEqn
LHS-0.7BE=RHS-0.7BE
14=0.3* BE
RearrangeEqn
Rearrange equation
0.3* BE=14
DivEqn
.LHS /0.3.=.RHS /0.3.
BE=14/0.3
UseCalc
Use a calculator
BE=46.66
After solving for BE, the length of BE can be added to CE to determine the length of the slope. The Slope's Length 20+46.66≈ 66.67 feet Dyland found that the length of the slope is approximately 66.67 feet.
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Discussion
Similarity of Slope Triangles
Slope triangles are right triangles that indicate the rise and run between two points on a line. The horizontal arrow denotes the run, while the vertical arrow represents the rise. Consider a pair of slope triangles on a line.
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Example
Comparing Slope Triangles
Later that day, Dylan returned home and realized that he still forgot about his math homework. He needs to finish it so that he can have free time on Saturday to go skateboarding again. His homework requires him to compare a pair of slope triangles that lie on the same line.
Consider that the side of each square in the grid is 1 unit. Help Dylan complete his homework.
a Write a proportion that compares the rise to the run for each of the similar slope triangles shown in the diagram by using the vertex labels.
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b Write the numeric value for the proportion found in Part A without simplifying fractions.
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Hint
a Write the ratio between corresponding sides of the two slope triangles to create a proportion. Then, rewrite the proportion so the ratio of the rise to the run of one triangle is on one side, and the ratio of the rise to the run of the other triangle is on the other side.
b Count the number of units from its starting point to its endpoint to determine the length of a specific segment.
Solution
a Consider that △ ABC and △ BDE are similar as they are slope triangles on the same line.
△ ABC~ △ BDE
Due to this similarity, the ratio between corresponding sides is equal. In this case, AC corresponds to BE and BC corresponds to DE. This information can be used to write the following proportion. AC/BE=BC/DE Since it is asked to compare the rise to the run for each of the similar slope triangles, this proportion will be rewritten to have the ratio of the rise to the run of △ ABC on one side and the ratio of the rise to the run of △ BDE on the other side.
AC/BE=BC/DE
MultEqn
LHS * BE=RHS* BE
AC=BC/DE* BE
MultEqn
LHS * DE=RHS* DE
AC * DE=BC* BE
DivEqn
.LHS /BC.=.RHS /BC.
AC * DE/BC=BE
DivEqn
.LHS /DE.=.RHS /DE.
AC/BC=BE/DE
This indicates that the ratios of the rise to the run are equal in both slope triangles.
b In the previous part, the proportion that compares the rise to the run for each of the similar slope triangles was written.
AC/BC=BE/DE The given diagram will be used to find the numerical value of this proportion. Count the number of units it takes to get from point A to point C to find the length of AC. Similarly, count the number of units it takes to get from point B to point C to find the length of BC.
This means that AC=3 and BC=2. Apply a similar reasoning to find the lengths of BE and DE.
Once of the lengths in the proportion are known, substitute them into the proportion to find its numerical value. AC/BC=BE/DE ⇓ 3/2=6/4
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Discussion
Slope Triangles and the Slope of a Line
The slope of a line is given by the ratio of the vertical change or rise
to the horizontal change or run
. This ratio remains equal across any two slope triangles formed by that line.
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Example
Slope of a Skate Park Shelter
At Dylan's skate park, there is a cool shelter that covers the entire skateboard track, providing shade for everyone during their skate sessions. Seeing the roof line, Dylan thinks it is the perfect chance to apply his new school knowledge about slope triangles on the same line.
With the help of a friend, Dylan measured the rise and run at various points on one of the slopes of the roof and sketched a diagram.
Now, he is curious whether the slope of the roof line remains consistent over the different areas. Help Dylan figure out the roof's slope and see if it remains the same everywhere.
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Hint
Start by choosing two points on the roof line and find the slope using the Slope Formula. Next, verify that the slope is the same by selecting a different set of points.
Solution
The slope of a line can be determined by substituting the coordinates of two points (x_1,y_1) and (x_2,y_2) into the Slope Formula.
m = y_2-y_1/x_2-x_1
In this case, ( 0, 3.9) and ( 2, 4.46) are two points on the roof line. Substitute these points into the formula to determine the roof's slope.
m = y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( 2,4.46) & ( 0,3.9)
m = 4.46- 3.9/2- 0
SubTerms
Subtract terms
m=0.56/2
CalcQuot
Calculate quotient
m=0.28
The slope of the line is m= 0.28. A different set of points can be used to confirm that the slope of the line remains the same everywhere. In this case, the points ( 4, 5.02) and ( 6.25, 5.65) can be used to evaluate the Slope Formula.
m = y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( 6.25,5.65) & ( 4,5.05)
m = 5.65- 5.02/6.25- 4
SubTerms
Subtract terms
m=0.63/2.25
CalcQuot
Calculate quotient
m=0.28
This confirms that the slope of the line is constant no matter which pair of points are used to calculate it.
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Closure
Perimeter and Scale Factor
Dylan had a great time skateboarding today and now he is sitting down to draw two ramps he dreams of skating on someday. He names them Little Leap
and Giant Glide.
Dylan draws them as triangles that look similar on paper, but Giant Glide
is just a bigger version of Little Leap.
a What is the perimeter of the triangle formed by the
Little Leapramp?
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b What is the perimeter of the triangle formed by the
Giant Glideramp?
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c Is it true that the perimeter of the bigger triangle is equal to the perimeter of the smaller triangle multiplied by a scale factor?
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Hint
a Add up the lengths of the smaller ramp's triangle to determine its perimeter.
b Add up the lengths of the bigger ramp's triangle to determine its perimeter.
c Find the ratio between the corresponding side lengths of the bigger triangle and smaller triangle to find the scale factor.
Solution
a The smaller ramp is known as the
Little Leapramp. It is formed by the legs of a triangle measuring 18.75 feet and 11.25 feet, with a hypotenuse of 21.87 feet. Add up the length of all three sides to find the perimeter of the
Little Leapramp.
Perimeter 18.75+11.25+21.87=51.87feet The perimeter of this triangle is equal to 51.87 feet.
b The perimeter of the triangle formed by the
Giant Glideramp can be found in a similar way. The legs of the triangle are of lengths 75 and 45 feet, respectively, while the hypotenuse has a length of 87.48 feet. Add these lengths to determine the perimeter of the triangle.
Perimeter 75+45+87.48=207.48feet
The perimeter of the triangle formed by the Giant Glide
ramp is 207.48 feet.
c Start by calculating the ratio between the corresponding side lengths of the triangles. Then, check if there exists the same scale factor between the perimeters of the triangles. In this case, use the base of the triangles, which measure 75 feet and 18.75 feet, respectively.
Scale Factor 75/18.75= 4 The scale factor is 4. Recall that the perimeter of the smaller triangle is 51.87 feet, and that of the bigger triangle is 207.48 feet. Now, calculate the ratio between the perimeters of the triangles. 207.48/51.87= 4 It can be concluded that the scale factors between the corresponding side lengths and perimeters of the triangles are the same. This refers that if the perimeter of the smaller triangle is multiplied by the scale factor, it equals to the perimeter of the bigger triangle. 4 * 51.87=207.48 Therefore, the given statement is true. If two figures are similar, then the ratio of their perimeters is equal to the scale factor.
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Similar Triangles
Exercise 3.1
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Let's start by considering a line with a slope of -2.
We are given that the rise of a slope triangle on this line is -4. We need to determine the run of this slope triangle. To begin, we will select a lattice point on the line as our starting point.
Next, we will draw the leg of the triangle, moving down 4 units since our rise value is negative.
We know that the ratio of the rise to the run of a slope triangle is equal to the slope of the line. rise/run = slope The rise of our slope triangle is -4 and the slope of our line is - 2. This allows us to calculate the corresponding run value for our rise.
The run of this slope triangle is 2 units. We can confirm this by looking at the graph.
Let's follow a similar process as in Part A) to find the run for a slope triangle with a rise value of -5. We will select a lattice point on the line as our starting point and then move 5 units down since the rise is negative.
Since we know that the ratio of the rise to run of a slope triangle is equal to the slope of the line, we will use the slope of -2 and the rise of -5 to find the run that corresponds to this rise.
The run of this slope triangle is 2.5 units. Let's confirm this by looking at the graph.
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Hint & Answer
S
Solution
Consider the following triangles.
Which of the following statements is true? Justify the answer.
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We will start by finding the values of x and y by creating a pair of equations using the Interior Angles Theorem. After that, we will substitute the values of x and y into each angle expression to calculate the angle measures. Finally, we will check if corresponding angles are equal to determine if the triangles are similar. Let's start by calculating the value of x.
Finding the Value of x
Consider the given diagram.
The Interior Angles Theorem states that the sum of the interior angles of a triangle equals 180^(∘). With this, we can write an equation for x.
Now, let's solve this equation to find the value of x.
Finding the Value of y
We will find the value of y by using a similar reasoning. According to the Interior Angles Theorem, the sum of the interior angles of this triangle equals 180^(∘). Let's use this to write an equation for y.
Let's solve this equation to find the value of y.
Checking Similarity
We need to determine if the two triangles are similar. We will first calculate the measures of all the angles in each triangle. Since we know that x= 7.5, we can substitute 7.5 for x in the expressions 12x and 6x to find the angle measures in the left triangle. cc 12x & 6x ⇓ & ⇓ 12( 7.5) & 6( 7.5) ⇓ & ⇓ 90 & 45 Similarly, we can substitute 22.5 for y in the expressions 4y and 2y to find the angle measures in the right triangle. cc 4y & 2y ⇓ & ⇓ 4( 22.5) & 2( 22.5) ⇓ & ⇓ 90 & 45 Once we have found the angle measures for both triangles, we can add this information into the diagram.
We can see that the angles in the left triangle are congruent to corresponding angles in the right triangle. Therefore, we can conclude that the two triangles are similar.
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Hint & Answer
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Solution
Similar Triangles
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