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
Exercise 2.1
Consider the following triangles.
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We are asked to determine which of the following statements are true. & △ XYZ ~ △ ABC & △ CBA ~ △ QRS & △ SQR ~ △ CAB & △ XZY ~ △ RSQ Let's consider the given diagram.
Our first step will be to calculate the measurements of the unknown angles. Once we have that information, we can move on to determine which statements from the list are true.
Finding Measures of the Angles
Let's start with △ ABC.
According to the Interior Angles Theorem, the sum of the measures of the interior angles of a triangle is always 180^(∘). This means that we can find the measure of m∠ B by subtracting m∠ A and m∠ C from 180^(∘). m∠ B=180^(∘) - 59^(∘) - 50^(∘) ⇓ m∠ B= 71^(∘) We will use the same method to determine the measures of m∠ Z and m∠ Q for △ XYZ and △ QRS.
In both of these triangles, the sum of the measures of the interior angles is 180^(∘).
| △ XYZ | △ RSQ |
|---|---|
| m∠ Z = 180^(∘) - 59^(∘) -71 ^(∘) | m∠ Q = 180^(∘) - 71^(∘) -50 ^(∘) |
| m∠ Z = 50^(∘) | m∠ Q = 59^(∘) |
We now know the measures of all the angles of the given triangles.
We can move on to verify the given statements.
Checking the Given Statements
We will evaluate the given statements by recalling that in similar triangles, corresponding angles are congruent. Let's look at the statements one by one, starting with △ XYZ~△ ABC.
△ XYZ~△ ABC
In this case, the pairs of corresponding angles are ∠ X and ∠ A, ∠ Y and ∠ B, and ∠ Z and ∠ C. △ X Y Z ~ △ A B C Now, let's take a look at these two triangles.
We can see that all the corresponding angles have equal measures, which means they are congruent. Therefore, the statement △ XYZ~△ ABC is true.
△ CBA~ △ QRS
Now, let's look at the second statement, △ CBA~△ QRS. In this case, the pair of corresponding angles are ∠ C and ∠ Q, ∠ B and ∠ R, and ∠ A and ∠ S. △ C B A ~ △ Q R S Let's look at the two triangles.
We can see that m∠ C is not equal to m∠ Q. This means that ∠ C and ∠ Q are not congruent. Since they are corresponding angles, this means that the statement △ CBA~△ QRS is not true.
△ SQR~△ CAB
Moving on to the third statement, △ SQR~△ CAB. In this case, the pairs of corresponding angles are ∠ S and ∠ C, ∠ Q and ∠ A, and ∠ R and ∠ B. △ S Q R ~ △ C A B Let's take a look at these two triangles.
We can see that all the corresponding angles have equal measures, which means they are congruent. Therefore, the statement △ SQR~△ CAB is true.
△ XZY~ △ RSQ
Finally, let's look at the fourth statement, △ XZY~△ RSQ. In this case, the pair of corresponding angles are ∠ X and ∠ R, ∠ Z and ∠ S and ∠ Y and ∠ Q. △ X Z Y ~ △ R S Q Let's look at the two triangles.
We can see that m∠ X is not equal to m∠ R. This means that ∠ X and ∠ R are not congruent. Since they are corresponding angles, this means that the statement △ XZY~△ RSQ is not true.
Conclusion
Finally, we can choose the true statements. ✓ & △ XYZ ~ △ ABC & △ CBA ~ △ QRS ✓ & △ SQR ~ △ CAB & △ XZY ~ △ RSQ
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Take a look at the following diagram.
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We need to find the distance across the river using the information in the given diagram. We will start by showing that △ CBA and △ DBE are similar. Once we establish that, we can use the proportional relationship between similar triangles to determine the missing distance.
We can see from the given diagram that both ∠ A and ∠ E are right angles, making them congruent as they have the same measure. We can also see that ∠ CBA and ∠ DBE are vertical angles, which means they are also congruent. Let's add this information into our diagram.
We have two pairs of congruent angles between △ CBA and △ DBE. Therefore, the triangles are similar according to the Angle-Angle Similarity Theorem. △ CBA ~ △ DBE Now, we can see that BE corresponds to BA and AC corresponds to ED.
Remember that corresponding sides of similar figures will have proportional lengths. We know that BE= 400 meters, BA= 120 meters, ED= 300 meters and AC=x. We can use these lengths to write a proportion. BA/BE=AC/ED ⇔ 120/400=x/300 Let's solve this proportion to find the value of x, which corresponds to the distance across the river.
Therefore, the distance across the river is 90 meters.
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A person stands 24 feet from a lamp post and casts a shadow as shown in the diagram.
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We are told that a person is standing 24 feet away from a lamp. We need to find out how much taller the lamp is than the person. We will use the given diagram that shows two similar triangles formed by the person, the lamp, and their shadows to find the answer to this question. Let's use h to represent the height of the lamp.
Because the triangles formed are similar, the corresponding sides of the triangles are proportional. Therefore, we can set up a proportion to find the height of the lamp. h/5.5=24+12/12 Let's solve this proportion to find the height of the lamp.
The height of the lamp is 16.5 feet. With this information, we can determine how many times taller the lamp is than the person by dividing the height of the lamp by the height of the person. 16.5/5.5=3 This means that the lamp is three times taller than the person.
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At an amusement park, there is a Ferris wheel that stands out as one of the tallest in the region, measuring approximately 41.5 meters in height. This Ferris wheel casts a shadow of about 20 meters. Suppose a person nearby has a shadow that extends to roughly 0.8 meters. Determine and calculate a proportion to estimate the height of this person.
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We know that the Ferris wheel is 41.5 meters tall and casts a shadow that is 20 meters long. In addition, a person standing nearby casts a shadow that is approximately 0.8 meters long. Let's draw a diagram that shows the situation to help us better understand this problem.
This is a shadow reckoning problem, so we assume that the angles formed by the Sun's rays when they hit the objects are congruent. The shapes created by the Sun's rays are right triangles.
If we think of the ground as a transversal and the Sun's rays as parallel lines, the angles created are corresponding angles. Therefore, the two triangles are similar by the Angle-Angle Similarity Theorem. This means the ratio of the heights of the Ferris wheel and the person is equal to the ratio of the lengths of their shadows. Height of the Giant Wheel/Height of the Man = Shadow of the Giant Wheel/Shadow of the Man ⇓ 41.5/h=20/0.8 Let's solve this proportion to find the value of h, which corresponds to the height of the person.
The person is 1.66 meters tall.
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Ramsha is calculating the height of a lighthouse based on the diagram and using the principles of similar triangles.
Below are the steps she took.
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Ramsha is finding the height of the lighthouse shown in the diagram.
Here are Ramsha's calculations, which we know are incorrect. We will retrace her steps to find and correct her mistake.
Let's assume that both the lighthouse and the house are at a right angle to the ground. Also notice that the angles formed by the Sun's rays when they hit the objects are congruent. This creates two similar triangles by the Angle-Angle Similarity Theorem.
This implies that the ratio of the distances from the objects to the points where the Sun rays hit the ground is equal to the ratio of the heights of the objects. House_d/Lighthouse_d & =House_h/Lighthouse_h &⇕ House_d/House_h & =Lighthouse_d/Lighthouse_h We will now substitute the height and distance values in the diagram into the proportion. House_d/House_h & =Lighthouse_d/Lighthouse_h &⇓ 33/66 & =99/x Now, let's review Ramsha's calculations. We see that the numerator and denominator of the fraction on the right are swapped. According to Ramsha, the ratio of the distance of the house to its height is equal to the ratio of the lighthouse's distance to its height. However, this statement is incorrect because these sides do not correspond with each other.
Therefore, Step 1 contains a mistake. Finally, let's correct Ramsha's mistake and solve the proportion we wrote to find the right height of the lighthouse.
Therefore, we conclude that the height of the lighthouse is 198 feet.
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