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Many people enjoy collecting miniature versions of famous characters or historical structures. These tiny copies are scale models of the originals — in other words, the model and the original figure are similar. This lesson will cover the characteristics of similar figures and explain how to calculate their measurements based on a specific scale.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Tadeo is interested in making miniature models of various objects. He wants to know more about how the dimensions of similar figures change depending on the scale. Consider the applet below, which displays two similar triangles.

Tadeo changes the length scale factor (lsf) and examines how the area of the blue triangle changes.

- What conclusions could Tadeo draw from the applet?
- How are the scale factor and the ratio between the areas of the triangles related?

Discussion

Two polygons are said to be similar if their corresponding sides are proportional and their corresponding angles are congruent. Because of this, there is a relation between the perimeters of similar polygons.

Rule

If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths.

Let $P_{1}$ and $P_{2}$ be the perimeters of $QRST$ and $ABCD,$ respectively. Let $ba $ be the scale factor between corresponding side lengths. Then, based on the above diagram, the following relation holds true.

$ABCD∼QRST⇒P_{2}P_{1} =ba $

Let $QRST$ and $ABCD$ be two similar polygons with perimeters $P_{1}$ and $P_{2},$ respectively. By the definition of similar polygons, corresponding side lengths are proportional, with the scale factor $ba $ as the common ratio.
Finally, the Perimeters of Similar Polygons Theorem is obtained by dividing both sides by $P_{2}$.

$⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧ ABQR =ba BCRS =ba CDST =ba DATQ =ba ⇔⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧ QR=ba ⋅ABRS=ba ⋅BCST=ba ⋅CDTQ=ba ⋅DA $

If the equations are added together, the left-hand side of the resulting equation will give the perimeter of $QRST.$ The right-hand side will give the scale factor times the perimeter of $ABCD.$
$QR+RS+ST+TQ=ba ⋅AB+ba ⋅BC+ba ⋅CD+ba ⋅DA$

FactorOut

Factor out $ba $

$QR+RS+ST+TQ=ba (AB+BC+CD+DA)$

SubstituteII

$QR+RS+ST+TQ=P_{1}$, $AB+BC+CD+DA=P_{2}$

$P_{1}=ba ⋅P_{2}$

$P_{1}=ba ⋅P_{2}⇔P_{2}P_{1} =ba $

Example

Tadeo likes playing basketball. He decides to make a miniature model of a basketball court with a length of $32$ centimeters.

External credits: freepik

A standard basketball court has a length of $28$ meters and a width of $15$ meters.

a Calculate the scale factor in this model. Write the answer in the simplest fraction form.

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"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:0.759765625em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">About<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5478515625em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">cm<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["98"]}}

a The miniature model of the basketball court will be a scale model of a real court, so the model and its original can be thought of as similar polygons. Therefore, their side lengths will be proportional and equal to the scale factor.

$Scale factor=Actual lengthLength of model $

Before the ratio can be calculated, both lengths need to have the same units of measure. The length of the real basketball court is $28$ meters and the length of the scale model is $32$ centimeters. Convert meters to centimeters to have same units of measure.
$28m⋅1m100 cm =2800cm $

Next, calculate the ratio of the length of the model to the corresponding actual length in centimeters.
$Scale factor=Actual lengthLength of model $

SubstituteValues

Substitute values

$Scale factor=280032 $

ReduceFrac

$ba =b/16a/16 $

$Scale factor=1752 $

b To calculate the perimeter of the model, first find the perimeter of the real basketball court.

$P=2(15+28)=86m $

The perimeter of the real basketball court is $86$ meters. Since the miniature model of the court and the real court are similar rectangles, the ratio of their perimeters is equal to the ratio of their corresponding side lengths, or the scale factor.
$P_{2}P_{1} =Scale Factor $

The scale factor is $1752 .$ Substitute the scale factor and the perimeter of the real basketball court into the equation and solve for $P_{1}.$
$P_{2}P_{1} =Scale factor$

SubstituteII

$P_{2}=86$, $Scale factor=1752 $

$86P_{1} =1752 $

MultEqn

$LHS⋅86=RHS⋅86$

$P_{1}=1752 ⋅86$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$P_{1}=175172 $

CalcQuot

Calculate quotient

$P_{1}=0.982857…$

RoundDec

Round to $2$ decimal place(s)

$P_{1}≈0.98$

$0.98m⋅1m100 cm =98cm $

The model basketball court has a perimeter of about $98$ centimeters.
Discussion

Like with perimeters, there is a relation between the areas of similar polygons.

If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Let $KLMN$ and $PQRS$ be similar figures, and $A_{1}$ and $A_{2}$ be their respective areas. The length scale factor between corresponding side lengths is $ba .$ Here, the following conditional statement holds true.

$KLMN∼PQRS⇒A_{2}A_{1} =(ba )_{2}$

The statement will be proven for similar rectangles, but this proof can be adapted for other similar figures.

The area of a rectangle is the product of its length and its width.

Area of $KLMN$ | Area of $PQRS$ |
---|---|

$A_{1}=KL⋅LM$ | $A_{2}=PQ⋅QR$ |

$⎩⎪⎪⎪⎨⎪⎪⎪⎧ PQKL =ba QRLM =ba ⇔⎩⎪⎪⎨⎪⎪⎧ KL=PQ⋅ba LM=QR⋅ba $

The next step is to substitute the expressions for $KL$ and $LM$ into the formula for $A_{1},$ which represents the area of $KLMN.$ $A_{1}=KL⋅LM$

SubstituteII

$KL=PQ⋅ba $, $LM=QR⋅ba $

$A_{1}=(PQ⋅ba )(QR⋅ba )$

▼

Simplify right-hand side

RemovePar

Remove parentheses

$A_{1}=PQ⋅ba ⋅QR⋅ba $

CommutativePropMult

Commutative Property of Multiplication

$A_{1}=ba ⋅ba ⋅PQ⋅QR$

ProdToPowTwoFac

$a⋅a=a_{2}$

$A_{1}=(ba )_{2}⋅PQ⋅QR$

AssociativePropMult

Associative Property of Multiplication

$A_{1}=(ba )_{2}(PQ⋅QR)$

$A_{1}=(ba )_{2}(PQ⋅QR)$

Substitute

$PQ⋅QR=A_{2}$

$A_{1}=(ba )_{2}A_{2}$

DivEqn

$LHS/A_{2}=RHS/A_{2}$

$A_{2}A_{1} =(ba )_{2}$

$Scale Factor ba ⇒ Area Scale Factor A_{2}A_{1} =(ba )_{2} $

Example

Now Tadeo wants to build a miniature spectator stand for his miniature basketball stadium. He plans to use the following stand as a model.
When viewed from the side, the stand looks like a right triangle. Tadeo knows that its hypotenuse is $5$ meters long and its height is $3$ meters.
### Solution

The scale model of the stand and the real stand have to be similar figures, so they have the same ratio between their corresponding side lengths. This ratio is called the scale factor.
The scale factor of the model will be $251 ,$ or $1:25.$ This means that the dimensions of the model stand will be $25$ times smaller than the dimensions of the real stand.
Since it is a right triangle, the Pythagorean theorem can be used to find $x.$
The base length of the triangular figure is $4$ meters. Now, calculate the area of the triangle as half of the product of the base and height.
The area of the triangular side of the real stand is $6$ square meters. Recall that if two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
The area of the model's triangular side is $0.0096$ square meters. Finally, convert it to square centimeters.

External credits: macrovector

a Tadeo makes his scale model stand with a height of $12$ centimeters. Calculate the scale factor in this model. Write the answer in the simplest fraction form.

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b Calculate the triangular side area of the miniature spectator stand.

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a The real spectator stand looks like a right triangle from a side perspective.

$Scale factor=Actual lengthLength of model $

Tadeo wants to create the model with a $12-$centimeter height. He knows that the height of the real stand is $3$ meters. Before finding the scale factor, both lengths must be in the same units. To achieve this, convert the height of the real stands from meters to centimeters.
$3m⋅1m100 cm =300cm $

Now the ratio of the height of the model to the corresponding actual height can be calculated.
$Scale factor=Actual lengthLength of model $

SubstituteValues

Substitute values

$Scale factor=30012 $

ReduceFrac

$ba =b/12a/12 $

$Scale factor=251 $

b Before calculating the side area of the model spectator stand, calculate the side area of the actual stand. This requires finding the length of the other leg.

$x_{2}+y_{2}=z_{2}$

SubstituteII

$y=3$, $z=5$

$x_{2}+(3)_{2}=(5)_{2}$

▼

Solve for $x$

CalcPow

Calculate power

$x_{2}+9=25$

SubEqn

$LHS−9=RHS−9$

$x_{2}=16$

SqrtEqn

$LHS =RHS $

$x_{2} =16 $

SqrtPowToNumber

$a_{2} =a$

$x=16 $

CalcPow

Calculate power

$x=4$

$A=21 bh$

SubstituteII

$b=4$, $h=3$

$A=21 ⋅4⋅3$

Multiply

Multiply

$A=21 ⋅12$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$A=212 $

CalcQuot

Calculate quotient

$A=6$

$A_{2}A_{1} =(ℓ_{2}ℓ_{1} )_{2}=k_{2} $

The ratio of the corresponding side lengths is equal to the length scale factor, so in this case, $k=251 .$ The square of this scale factor can be used to find the area of the model's triangular side.
$A_{2}A_{1} =k_{2}$

SubstituteII

$A_{2}=6$, $k=251 $

$6A_{1} =(251 )_{2}$

▼

Solve for $A_{1}$

PowQuot

$(ba )_{m}=b_{m}a_{m} $

$6A_{1} =25_{2}1_{2} $

CalcPow

Calculate power

$6A_{1} =6251 $

MultEqn

$LHS⋅6=RHS⋅6$

$A_{1}=6251 ⋅6$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$A_{1}=6256 $

CalcQuot

Calculate quotient

$A_{1}=0.0096$

$0.0096m_{2}⋅1m_{2}10000 cm_{2} =96cm_{2} $

The area of the triangular side of Tadeo's model is $96$ square centimeters.
Pop Quiz

Consider two similar polygons. Use the given information to find the scale factor, perimeter, or area of either of the polygons. Keep in mind that the given length scale factor corresponds to the ratio of Polygon $2$ to Polygon $1.$ Round the answer to two decimal places if necessary.