Book I.  Proposition 20

Problems

1.   a)  State the hypothesis of Proposition 20.

These are any two sides of a triangle.

1.  b)  State the conclusion.

Together they will be greater than the third side.

1.  c)  Practice Proposition 20.

  2.   In an equilateral triangle, the sides are in the ratio 1 : 1 : 1; that is, they are equal to one another.  How does that illustrate Proposition 20?
 
  1 + 1 is greater than 1.

3.  a)  Can an isosceles triangle have sides in the ratio 1 : 1 : 2?

No. The equal sides  1 + 1  are not greater than 2.

  3.  b)   Can an isosceles triangle ever have sides in the ratio of natural numbers?
 
  Yes, if the unequal side is less than the equal sides.  For example, 2 : 2 : 1.
  4.   Let the perimeter of a scalene triangle be a natural number of units. What is the smallest perimeter such that the sides will be in the ratio of natural numbers?

2 : 3 : 4

5.   A scalene triangle has one side that is 2 cm.  Can the remaining sides
5.   be multiples of 2 cm?


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