Geometry: An Interactive Journey to Mastery

Solutions

Lesson 28

1. On a square grid, mark a point O, and mark a point P 1 unit to the east and 3 units to the north. Use OP as
   a radius to draw a circle with center O. In this picture, we see a 2 × 6 rectangle and a square of side length
   20. The ratio of their areas is^1220 35.


2. 7KLVLVDYHU\GLI¿FXOWTXHVWLRQ
   The following is the picture of a tetrahedron with the midpoints
   of its sides marked and line segments connecting drawn to create
   DQLQQHU³PLGSRLQW¿JXUH ́௘௘6HHFigure S.28.1௘ͽ
   :HVHHWKDWWKHLQQHU¿JXUHLVDUHJXODURFWDKHGURQ
   This shows that one octahedron and one tetrahedron stack together to
   PDNHD¿JXUHZLWKXQH[SHFWHGÀDWIDFHV7KHUHDUHÀDWIDFHVLQDOO


3. 6XSSRVHWKHYROXPHRIWKHODUJHWHWUDKHGURQLQWKHVROXWLRQWR3UREOHPLVV.
   Each small tetrahedron in the corner of this large one is a scaled copy with scale factor k 12.
   Thus, its volume is kV^3 V 8. This means that the volume of the interior octahedron is Vu 4.VV 82
   The ratio of volumes of one small tetrahedron to the octahedron is^814.
   2

V

V^

4. /DEHODQJOHw and sides a and b as shown in Figure S.28.2.
   Notice that PQ 1.x^2
   Looking at the small shaded triangle and the large shaded
   triangle, we see

(^)
2
2
sin 1
cos 11.
w axx x
w bx x
(^) 
(^) 
6Rab 11 xx^2 xx 22 and.
Figure S.28.1
x
x
P x

Q

x

b a

a

w

b

íx

íx

íx

íx

Figure S.28.2