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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

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

Pop Quiz

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

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.
In similar triangles, the ratios between corresponding sides are equal. This means that the ratio of the lengths of the shadows of the flagpoles equals the ratio of the lengths of the flag poles. The following proportion can then be set.

$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

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.

Dylan positioned himself along the tree shadow so that the transversal from the top of the tree to the end of its shadow formed two similar triangles. Using the information provided in the diagram, help Dylan calculate the tree's height.

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The ratio of the lengths of corresponding sides of similar triangles is always the same.

Begin by labeling the vertices for each of the triangles in the diagram.
Dylan determined that the height of the tree is $39.5$ feet.

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.

$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 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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mrel\">\u2248<\/span><\/span><\/span><\/span>","formTextAfter":"feet","answer":{"text":["66.67"]}}

The ratio of the lengths of corresponding sides of similar triangles is always the same.

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 $0.70.$
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=10.5ft $

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 $BC.$ $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≈66.67feet $

Dyland found that the length of the slope is approximately $66.67$ feet. Discussion

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

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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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.

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 $DE.$

Once of the lengths in the proportion are known, substitute them into the proportion to find its numerical value.$BCAC =DEBE ⇓23 =46 $

Discussion

The slope of a line is given by the ratio of the vertical change or

riseto the horizontal change or

run. This ratio remains equal across any two slope triangles formed by that line.

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 — $(x_{1},y_{1})$ and $(x_{2},y_{2})$ — on the line are known, the Slope Formula can be used.

$m=x_{2}−x_{1}y_{2}−y_{1} $

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

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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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.

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.
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=x_{2}−x_{1}y_{2}−y_{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=x_{2}−x_{1}y_{2}−y_{1} $

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=x_{2}−x_{1}y_{2}−y_{1} $

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

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
### Hint

### Solution

*true*. If two figures are similar, then the ratio of their perimeters is equal to the scale factor.

Little Leapand

Giant Glide.Dylan draws them as triangles that look similar on paper, but

Giant Glideis just a bigger version of

Little Leap.

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.

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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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.

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=51.87feet $

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.

$Perimeter75+45+87.48=207.48feet $

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 Loading content