August 7, 2002

Similar Objects

  1. Corresponding angles are congruent.
  2. Corresponding sides are proportional.
    1. Ratios of corresponding sides are the same.

If you scale a polygon by ‘s’, then the area is scaled by what factor?

  1. The area is scaled by the square of scaling factor. 

In the right triangle, if you draw the altitude to the hypotenuse, then the triangles formed are similar to the original.  In particular the hypotenuses are c, a & b.  This makes the scaling factors from the original triangle (w/ c as the hypotenuse) to the other two (w/ a & b as hypotenuse) are:

C                     A                     B

Lengths            1                      A/C                  B/C

Area                 1                      (A/C)2              (B/C)2

So if the sum of the two smaller triangles is equal to the area of the larger triangle then:

1 = (A/C)2  + (B/C)2

So if we multiply both sides by C2, then (surprise!!),

C2 = A2 + B2

Now, build a cube out of the polydrons. 

What card will you need to add to divide the cube in half?  (Diagonally…)  What is the length the card will need to be?  ( )

What is length of the diagonal of the entire cube?  ( )

If I want to make a pyramid inside the cube that will have a square base but with the vertex at one of the vertices of the cube opposite the base, what would the dimensions be?

Sides                Volume of Prism                       Volume of pyramid

SxSxS                          S3                                             S3/3

SxSxS                          S2H                                          S2H/3

Bxh                              Bh                                            Bh/3