Finding Values

Finding Values

Jan. 27 - 28

Learning Goals

Use similarity, proportionality, and the Pythagorean Theorem to solve problems involving right triangles

Warm - up

5 minutes

Identify the picture that you feel doesn't belong, and explain why

Open Ended

Which one doesn't belong? Why?

Missing Information

Using the given information, fill in the missing side lengths of the desired triangles

Don't get caught on extraneous (unnecessary) information. Think about what it is you need to know in order to solve the problem

Given the following, find XY, PR, and QR:

•  $\Delta XYZ\ \sim\ \Delta PQR$  

•  $XZ=6,\ YZ=8$  
  

•  $PQ=15$  

•  $\angle Z\ and\ \angle R$  are right angles

•  $m\angle P=53\degree$  

•  $m\angle Y=37\degree$  

• Scale factor from  $\Delta PQR\ to\ \Delta XYZ$  is  $\frac{2}{3}$  

Open Ended

Find the values of PR, XY, and QR

Finding PR

• Create proportion

   $\frac{6}{PR}=\frac{2}{3}$  

• cross multiple $18=2\left(PR\right)$  

• Solve for PR by dividing by 2

•  $PR=9$  

Finding XY

• Create proportion

   $\frac{XY}{15}=\frac{2}{3}$  

• cross multiply  $30=3\left(XY\right)$  

• Divide by 3: XY = 10

Finding QR

• create proportion

   $\frac{8}{QR}=\frac{2}{3}$  

• cross multiply  $24=2\left(QR\right)$  

• Divide by 2: QR = 12

Since you had right triangles...

• You could also use Pythagorean Theorem to solve for the missing sides

•  $m\angle Z=90\degree$  means XZ and YZ are the legs

•  $XY=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$  

To find PR and QR:

• You know that

   $\Delta XYZ$   is a 6, 8, 10 right triangle

• You know PQR is 3/2 (or 1.5) times bigger than XYZ (the scale factor from PQR to XYZ is 2/3, which means knowing the smaller triangle you need to flip your fraction to work backwards to the big triangle)

• multiply your known sides by 1.5 to find PQR

•  $6\times1.5=9\ \rightarrow\ PR=9$  

•  $8\times1.5=12\ \rightarrow\ QR=12$  

Side Note: Mirrors

• In school we'd go outside with mirrors and estimate/measure trees, buildings, structures, etc

• Yesterday's practice problems asked about this (it was #4)

• All you need is your height, the distance you are from the mirror, and the distance the mirror is from your object

• Then use similarity to solve the height of the building

Finding triangles in other shapes

Such as rectangles

Open Ended

Find the value of x.

How'd you find it?

 $x=\sqrt{65}$  

Or other equivalent answer (If you used a decimal you cannot use the equal sign though...)

 $x\approx8.06$  
 $x\approx8.1$  

(Perks of keeping the radical you can use the equal sign)

Open Ended

Find the value of y.

How'd you find it?

 $y=\sqrt{260}=2\sqrt{65}$  

If you don't know/remember how to simplify radicals don't worry, we'll go over that a little later when we need it

 $y\approx16.12$  (or something similar)