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Similar Figures


(Not exactly the same,
  but pretty close!)
Let’s do a little
  review work
before discussing
 similar figures.
Congruent Figures
• In order to be congruent, two
  figures must be the same size
  and same shape.

                   ≅
Similar Figures
• Similar figures must be the
  same shape, but their sizes may
  be different.

                  ≅
Similar Figures
 ≅   This is the symbol that
 means “similar.”
These figures are the same
 shape but different sizes.

               ≅
SIZES
• Although the size of the two
  shapes can be different, the
  sizes of the two shapes4must
  differ by a factor.
      2
                    6       6
  3
      1
          3    ≅        2
SIZES
• In this case, the factor is x 2.

                           4

       2


   3       3
                ≅     6         6

       1                    2
SIZES
• Or you can think of the
  factor as     2.
                        4

       2


   3       3
               ≅    6           6

       1                    2
Enlargements
• When you have a photograph
  enlarged, you make a similar
  photograph.


         ≅
         X3
Reductions
• A photograph can also be
  shrunk to produce a slide.


                4

            ≅
Determine the length of the
          unknown side.

          15
12
               ≅     4
                             ?

                         3
      9
These triangles differ by a factor
                   of 3.

                    15       3= 5
             15
12                 ≅     4
                                  ?

                              3
        9
Determine the length of the
         unknown side.
                        ?


    2
4         ≅     24
These dodecagons differ by a
            factor of 6.
                         ?
                = 12
         2 x6
    2
4         ≅        24
Sometimes the factor between 2
     figures is not obvious and some
        calculations are necessary.


12             15
                     ≅    8         10

          18                   12
                    ?=
To find this missing factor,
          divide 18 by 12.



12           15
                   ≅     8          10

        18                    12
                  ?=
18 divided by 12
      = 1.5
The value of the missing
          factor is 1.5.


12          15    ≅      8        10

       18                    12
                 1.5 =
When changing the size of a
     figure, will the angles of the
         figure also change?

                               ?
     40



70        70              ?           ?
Nope! Remember, the sum of all 3
angles in a triangle MUST add to 180
                degrees.
           If the size of the
              angles were        40
               increased,
    40          the sum
             would exceed
                  180
                degrees.
70      70                70           70
We can verify this fact by placing
 the smaller triangle inside the
        larger triangle.

                            40
     40



70        70           70        70
The 40 degree angles
are congruent.
                            40



                       70        70
                   70             70
The 70 degree angles
are congruent.

                                  40

                             40



                        70
                       70          70 70
The other 70 degree
angles are congruent.

                            4
                                40



                           70
                        7070         70
                                      70
Find the length of the missing
             side.

                          50
     30          ?
 6


                     40
                8
This looks messy. Let’s
translate the two triangles.

                         50
    30          ?
6


                    40
               8
Now “things” are easier to see.


                                50
             30
             ?
6

                           40
         8
The common factor between
these triangles
 is 5.
                           50
           30
           ?
6

                      40
       8
So the length of
the missing side
     is…?
That’s right! It’s ten!


                            50
        30
        10
6

                       40
    8
Similarity is used to answer real
         life questions.
       • Suppose that you
         wanted to find the
         height of this tree.
Unfortunately all that
  you have is a tape
measure, and you are
too short to reach the
   top of the tree.
You can measure the length of
     the tree’s shadow.




      10 feet
Then, measure the length of your
           shadow.




       10 feet             2 feet
If you know how tall you are,
then you can determine how tall
          the tree is.




                 6 ft
       10 feet            2 feet
The tree must be 30 ft tall. Boy,
        that’s a tall tree!




                   6 ft
       10 feet              2 feet
Similar figures “work” just like
     equivalent fractions.


 30                5
 66               11
These numerators and
denominators differ by a factor of
               6.

  30           6
                    5
               6
  66               11
Two equivalent fractions are
   called a proportion.


30               5
66              11
Similar Figures

• So, similar figures are
  two figures that are the
  same shape and whose
  sides are proportional.
Practice Time!
1) Determine the missing side of
             the triangle.

                             ?
         5          9
3               
        4                   12
1) Determine the missing side of
             the triangle.

                             15
         5          9
3               
        4                   12
2) Determine the missing side of
         the triangle.


6       6          36        36
               
    4
                         ?
2) Determine the missing side of
         the triangle.


6       6          36         36
               
    4
                         24
3) Determine the missing sides of
          the triangle.


 39                   ?
         33                ?
                      8
24
3) Determine the missing sides of
          the triangle.


 39                   13
         33                11
                      8
24
4) Determine the height of the
             lighthouse.


8                ?

      2.5
                          10
4) Determine the height of the
             lighthouse.


8                32

      2.5
                          10
5) Determine the height of the
                car.


                  ?
3             
       5                    12
5) Determine the height of the
                car.


                  7.2
3             
       5                    12
THE END!

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7 - similar figures