Entry 7:

Some common uses of similar triangles in real life range from nearly anything you can imagine, from trusses on houses, to sails on a sail boat. you can use this anytime something arises, and not just with triangles either! Let's say you own a sailboat,

Nice boat! but, lets say you need to buy a new sail,  the one you have ripped. So you order a new one online, but you get it, and it's too big!

Instead of returning it, because the company won't let you, you decide to cut it down to the proper size.

So, lets say that your sail is  6 feet tall, and 4 feet wide at the bottom.

So you take your sail, and measure a 4 foot wide base, and mark it, and go up on the altitude of the triangle six feet snd mark it. You knowbthat both triangles are righht triangles, and have an angle of 45 degrees. You make a lin going from the corner, to the six foot mark and cut it. that ins how you use similar triangles in a real life scenario!!

tips from pros unit 1

unit 1 day 8

Definition of Scale Factor

• The ratio of any two corresponding lengths in two similar geometric figures is called as Scale Factor.
• The ratio of the length of the scale drawing to the corresponding length of the actual object is called as Scale Factor

scale factor is where you want to find how much bigger the other object is and you do that by taking the bigger object number angle and dividing it by the smaller one such 

′
 once you divide it out, you get 1.5 and that means that the newer object is 1.5 times larger than the original.




Here's a practice quiz!

unit 1 day 6

Here, we are talking to you about angle angle. Angle angle is where you take 2 triangles and see if they're similar.

We do that by, taking 2 triangles--

Now we see that they are corespondent, a-e  b-g  c-f

lets say a-e are 75

and b-g are 69

now we have to find c-f

in order to do so, we take 180(because the total sum of a triangle's angles are 180) and then we take 180-75-69 and we end up with 36. that means that c and f are 36. When you add them up, 75+69+36 then you would get 180, in other words, you subtract 2 angles from 180 to get the third, and if both triangles have 2 equal degree of sides, then they are similar.

Here is a little practice quiz :

Is triangle PQR congruent to
            triangle STV by SAS? Explain.

Show that triangle BAP is congruent
              to triangle CDP.

  Problem 1Segment PQ is congruent
              to segment ST because
              PQ = ST = 4.
            Angle Q is congruent to
              angle T because
              angle Q = angle T = 100 degrees.
            Segment QR is congruent
              to segment TV because QR = TV = 5.
            Triangle PQR is congruent 

              

 Problem 2 Angle A is congruent to angle D
                  because they are both right angles.
               Segment AP is congruent to segment DP be-
                  cause both have measures of 5.
               Angle BPA and angle CPD are congruent be-
                  cause vertical angles are congruent.
               Triangle BAP is congruent to triangle CDP
                  by Angle-Side-Angle.

Unit 1:Entry 2
In this entry, we will go through what similar, and congruent shapes are, and the differences, and also how to find the area of similar/congruent shapes. We'll start by defining these terms, in "Math speak"

Similar Shapes:These are two or more shapes that are the same shape, just reduced or enlarged.

 Here is an example of two similar shapes.

Congruent shapes: these are two or more shapes that are exactly identical.


In this picture, all the side lengths on triangle ABC are equal to those on EDF. making them Congruent.

So, now that you have that out of the way, we're gonna work on finding the area of similar shapes, because congruent shapes are identical, their area/perimeter will be too!

For this example, lets use some similar triangles.
Similar triangles have the same shape, but they are scaled differently, so they may  be different sizes.
This is where you may need to use some base knowledge of proportions


In this example the smaller triangle, is exactly 1/2 as big as the large one, this is made prevalent by the length markings on each leg of the shape.

Lets move on to a more in depth example
 A= 1/2 4*12=24

A=1/28*24=96
Notice, that 24/96 reduces too 1/4. you can see here that triangle 1 is 1/4 the size, or 4 times smaller than triangle 2. this is due to the similarity ratio, or scale factor.
now, you may be asking, how do you find the ratio?
all you have to do is pick to corresponding sides, and set it up as if it were a fraction, and then simplify it!
It's super simple, just as I demonstrated in the above problem!



Lets try some practice problems!
The key will be below.



In this triangle, A'C' is paralell to AC.  Find the length of B'C and  A'A





















 A research team wishes to determine the altitude of a mountain as follows: They use a light source at L, mounted on a structure of height 2 meters, to shine a beam of light through the top of a pole P' through the top of the mountain M'. The height of the pole is 20 meters. The distance between the altitude of the mountain and the pole is 1000 meters. The distance between the pole and the laser is 10 meters. We assume that the light source mount, the pole and the altitude of the mountain are in the same plane. Find the altitude h of the mountain. 










The two triangles are similar and the ratio of the lengths of their sides is equal to k: AB / A'B' = BC / B'C' = CA / C'A' = k. Find the ratio BH / B'H' of the lengths of the altitudes of the two triangles.  (NOTE: The altitude of a triangle is the height)




























_______________________________K E Y_____________________________________






Problem one:
First, use the proportionality of the lengths, to write an equation that helps solve for X and Y
(30 + x) / 30 = 22 / 14 = (y + 15) / y

The quation for X may be written like: (30+x)/30=22/14 = (Y+15) /y

Then you solve the above for X
420 + 14 x = 660 

x = 17.1 (rounded to one decimal place). 

The equation for Y may be written like:
22/14 =(y+15)/y
THen you solve the above, to get the answer: Y=26.25









Problem Two: 
First draw a horizontal line LM. PP' and MM' are vertical to the ground and therefore parallel to each other. Since PP' and MM' are parallel, the triangles LPP' and LMM' are similar. Hence the proportionality of the sides gives: 

1010 / 10 = (h - 2) / 18 
Then solve for H to get,: H=1820 meteres.


Problem Three:
If the two triangles are similar, the corresponding angles are 100% congruent, therefore the angles BAH and B'A'H' are exactly the same. These triangles have two pairs of corresponding congruent anges, BAH and B'A'H' and the right triangles BHA and B'H'A'. they are similiar, therefore; AB/ A'B'=BH/B'H'=k

UNIT 1-day 4

For instance, say you were an engineer putting up glass on the Luxor hotel in LAS VEGAS.
It is a 3-d pyramid and you are the leader for how big the hotel is.

to find the total surface area, you are to take the base of the luxor. The base is 646ft. Then, you are to find the slant height which is 476ft. to find the total surface area,  it's 2bs+b2
so that means, you take the base length and multiply it by the slant height. after you do that, you multiply it by 2.
it should look like--
646ft x 476ft=307,496 x 2=614,992

after that, you take the base and square it. 646ft(2)  and that gets you__360,000

after the 2 numbers are acheived, you add them. 360,000+614,992= 974,992ft 2**total surface area***                                                            ^^^^^^



quiz

A-a pyramid has the base of 120ft and the height of 300ft, what is the surface area?
B- a cylinder has a height of 10ft and a radius of 50ft, what is the surface area?
C- a prism has a base of 300ft and height of 27ft, what is surface area?

answers



A-259,200ft2
B-17,584ft2
C-8100ft2

Entry:5
In thie entry, we're gonna touch on  how to find the missing lengths of a similar triangle. If you remember from entry 2, Similar triangles have the same shape, but they are scaled differently, so they may  be different sizes.
This is where you may need to use some base knowledge of proportions.
You can use proportions to figure out the side length.

If a triangle has a height of 12, length of 2 and hypoteneuse of 24
and a similar triangle has a height of 24, length of 4, but the  length of the hypotenuse isn't given, you can set up a ratio.
12/24
6/12
3/6
1/2
so the triangle is  1 half of the size as the larger one.
you can do this with any triangle, as long as at least one side length on each shape is given, then you can set up a ratio,  and work from there, getting the ratio, simplifying it and coming up with the fraction; whatever it may be


                                            Lets do a practice quiz!

On a sunny day, Michelle and Nancy noticed that their shadows were different lengths. Nancy measured Michelle's shadow and found that it was 96 inches long. Michelle then measured Nancy's shadow and found that it was 102 inches long.

a.	Who do you think is taller, Nancy or Michelle? Why?
b.	If Michelle is 5 feet 4 inches tall, how tall is Nancy?
c.	If Nancy is 5 feet 4 inches tall, how tall is Michelle?

Nancy is taller. Since the right triangles defined by their heights and their shadows are similar, then the bases of the triangles have to be proportional to the heights of the triangles (i.e., their body heights).

b. 

Converting Michelle's height into inches (64 inches) and setting up a proportion, you would have:

64 / x = 96 / 102, or

x = 68 

Converting 68 inches back to feet, Nancy is 5 feet 8 inches tall.

c. 

Converting Nancy's height into inches (64 inches) and setting up a proportion, you would have:

64 / x = 102 / 96, or

x = 60.24

Converting 60.24 inches back to feet, Michelle is approximately 5 feet and 1/4 inch tall.