4. Similar Triangles
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Chapter 11
4.
Similar Triangles
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.
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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:
Triangles and Measurement:
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.
Now, consider these questions.
- 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?
Discussion
Angle-Angle Similarity Theorem
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.
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?
15h=128
This proportion can be solved to determine the height of the small flagpole.
15h=128
MultEqn
LHS⋅15=RHS⋅15
h=128⋅15
MoveRightFacToNum
ca⋅b=ca⋅b
h=128⋅15
Multiply
Multiply
h=12120
CalcQuot
Calculate quotient
h=10
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.
Dylan determined that the height of the tree is 39.5 feet.
AEAD=CEBD
Note that AE equals the sum of AD and DE.
AD+DEAD=CEBD
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+DEAD=CEBD
SubstituteValues
Substitute values
13+6613=CE6.5
AddTerms
Add terms
7913=CE6.5
MultEqn
LHS⋅CE=RHS⋅CE
7913⋅CE=6.5
MultEqn
LHS⋅79=RHS⋅79
13⋅CE=513.5
DivEqn
LHS/13=RHS/13
CE=39.5
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.
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Hint
The ratio of the lengths of corresponding sides of similar triangles is always the same.
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.
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.
After solving for BE, the length of BE can be added to CE to determine the length of the slope.
Dylan’s Current Height0.70⋅15=10.5 ft
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. △ABC∼△DBE
Consequently, DE corresponds to AC and BE corresponds to BC. DEBE↔↔ACBC
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. ACDE=BCBE⇓ACDE=CE+BEBE
In this equation, only BE is unknown. The known values can be substituting in this proportion to find the value of BE.
ACDE=CE+BEBE
SubstituteValues
Substitute values
1510.5=20+BEBE
CalcQuot
Calculate quotient
0.7=20+BEBE
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=0.314
UseCalc
Use a calculator
BE=46.66
The Slope’s Length20+46.66≈66.67 feet
Dyland found that the length of the slope is approximately 66.67 feet. 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. 
Because slope triangles are right triangles, ∠BED and ∠ACB are congruent since they are both right angles. In addition, AC is parallel to BE and both are cut by the line, which is also a transversal. As a result, the corresponding angles ∠DBE and ∠BAC are also congruent. Therefore, the slope triangles are similar, as per the Angle-Angle Similarity Theorem.
∠ACB≅∠BED∠BAC≅∠DBE⇓△ABC∼△BDE
This fact holds for any pair of slope triangles on the same line.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
BEAC=DEBC
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. BEAC=DEBC
MultEqn
LHS⋅BE=RHS⋅BE
AC=DEBC⋅BE
MultEqn
LHS⋅DE=RHS⋅DE
AC⋅DE=BC⋅BE
DivEqn
LHS/BC=RHS/BC
BCAC⋅DE=BE
DivEqn
LHS/DE=RHS/DE
BCAC=DEBE
b In the previous part, the proportion that compares the rise to the run for each of the similar slope triangles was written.
BCAC=DEBE
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.
BCAC=DEBE⇓23=46
Discussion
Slope Triangles and the Slope of a Line
The slope of a line is given by the ratio of the vertical change or 
This shows that the line's steepness is constant, no matter which points are used to form the slope triangles. Also recall that when two points — (x1,y1) and (x2,y2) — on the line are known, the Slope Formula can be used.
riseto the horizontal change or
run. This ratio remains equal across any two slope triangles formed by that line.
m=x2−x1y2−y1
This formula directly applies the principle that the ratio of the rise to the run between any two points on a line remains constant.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 (x1,y1) and (x2,y2) into the Slope Formula.
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.
This confirms that the slope of the line is constant no matter which pair of points are used to calculate it.
m=x2−x1y2−y1
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=x2−x1y2−y1
SubstitutePoints
Substitute (2,4.46) & (0,3.9)
m=2−04.46−3.9
SubTerms
Subtract terms
m=20.56
CalcQuot
Calculate quotient
m=0.28
m=x2−x1y2−y1
SubstitutePoints
Substitute (6.25,5.65) & (4,5.05)
m=6.25−45.65−5.02
SubTerms
Subtract terms
m=2.250.63
CalcQuot
Calculate quotient
m=0.28
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 
He wonders if the way around the ramps, or their perimeters, are related. Help him find the following information to find out the answer to his question.
Little Leapand
Giant Glide.Dylan draws them as triangles that look similar on paper, but
Giant Glideis 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.
Perimeter18.75+11.25+21.87=51.87 feet
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.
Perimeter75+45+87.48=207.48 feet
The perimeter of the triangle formed by the Giant Glideramp 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 Factor18.7575=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.
51.87207.48=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.
Similar Triangles
Exercises
Are the pair of triangles similar?
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We will use the Angle-Angle Similarity Theorem to determine if the given triangles are similar. Let's first take a look at the given triangles.
We can see that ∠ A and ∠ D are congruent since they both measure 70^(∘). However, ∠ C and angle ∠ E are not congruent. That being said, ∠ B might still be congruent with ∠ E. We can determine the measure of ∠ B by using the Interior Angles Theorem, which states that the sum of all interior angles is 180^(∘). Interior Angles Theorem m∠ A+ m∠ B + m∠ C=180^(∘) Therefore, we can subtract m∠ A and m∠ C from 180^(∘) to find m∠ B. m∠ B=180^(∘)-70^(∘)-39^(∘) ⇓ m∠ B=71^(∘) This means that m∠ B is 71^(∘).
Now that we know m∠ B, we can see that ∠ B and ∠ E are congruent. Since we have two pair of congruent angles, the Angle-Angle Similarity Theorem tells us that the triangles are similar. ∠ A ≅ ∠ D ∠ B ≅ ∠ E ⇓ △ ABC ~ △ DEF
We will use a similar process to determine whether the triangles are similar. Start by examining the given triangles.
We have two angles, ∠ A and ∠ D, that are congruent. Let's see if we can find another pair of congruent angles by determining the measure of ∠ B. To do so, let's subtract m∠ A and m∠ C from 180^(∘). m∠ B=180^(∘) - 63^(∘) -55^(∘) ⇓ m ∠ B = 62^(∘) We can find m∠ F using a similar process. m∠ F=180^(∘) - 55^(∘) -55^(∘) ⇓ m ∠ F = 70^(∘) Therefore, m∠ B is 62^(∘) and m∠ F is 70^(∘).
Now that we know all the angle measures, we can see that there are no two pairs of congruent angles between these triangles. Therefore, we can conclude that the triangles are not similar.
Let's apply a similar fashion to find out if the following triangles are similar.
We can observe that ∠ C and ∠ F are congruent since they have the same measure. Next, we will check if we can identify another pair of congruent angles. To do so, we have to calculate m∠ A by subtracting m∠ B and m∠ C from 180^(∘). m∠ A=180^(∘) - 95^(∘) -47^(∘) ⇓ m ∠ A = 38^(∘) The measure of angle A is 38^(∘).
From the diagram, we can see that ∠ A is congruent to ∠ D. We have now found two pairs of congruent angles between △ ABC and △ DEF. This means that the two triangles are similar. ∠ A ≅ ∠ D ∠ C ≅ ∠ F ⇓ △ ABC ~ △ DEF
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Use the similarity between the pair of triangles to find the missing side length.
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We are told that △ ABC and △ DEF are similar triangles.
Because of the similarity, the ratio between corresponding sides is equal. In this case, AB corresponds to DE and AC corresponds to DF. We can use this information to write a proportion that relates the side lengths of the two triangles. AB/DE=AC/DF We have been given the values of AB, AC, and DE, so we can substitute these values into the proportion and solve for DF.
This means that DF equals 4.8 feet.
This time, we need to determine the value of EF. Let's begin by considering the given triangles.
Since the triangles are similar, we know that the ratio between corresponding sides is equal. In this case, DE corresponds to AB and EF corresponds to BC. Therefore, we can set up the following proportion. DE/AB=EF/BC We can substitute the values of DE, AB, and BC, and solve the proportion for EF.
Therefore, EF is 6 centimeters.
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Take a look at the diagram that shows the shadows cast by a 2-foot tall dog and a tree.
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Let's start by looking at the given diagram.
Assuming that the tree and the dog are perpendicular to the ground, they create right angles with the ground. In addition, the angles formed by the rays of the Sun when they hit each object and the ground are equal. This implies that there are two pairs of congruent angles, resulting in two similar triangles as per the Angle-Angle Similarity Theorem.
This means the ratio of the tree's height to the dog's height is equal to the ratio of the tree's shadow to the dog's shadow since the corresponding sides of similar triangles are proportional. Tree's Height/Dog's Height=Tree's Shadow/Dog's Shadow ⇓ h/2=25/5 Let's solve this proportion to find the value of h, which corresponds to the height of the tree.
Therefore, the height of the tree is 10 feet.
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Observe the diagram of the shadows cast on the ground at the same spot by a house and a school building.
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Let's begin by considering the given diagram.
Let's assume that the house and the school are positioned at right angles to the ground. Also, the angles created by the Sun's rays as they hit each object and the ground are equal. This results in two pairs of congruent angles, which satisfies the Angle-Angle Similarity Theorem and creates two similar triangles.
This means that the ratio of the school's height to the house's height equals the ratio of the school's shadow to the house's shadow. This is because the ratio between corresponding sides of similar triangles are equal. School's Height/House's Height=School's Shadow/House's Shadow ⇓ h/3=12/6 Let's solve this proportion to find the value of h.
Therefore, the height of the school is 6 meters.
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Similar Triangles
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