MATH 3181 - 001 FALL 1996

Homework Assignment 6
Due Monday, December 2, 1996

  1. Write down equations which show how the side and the angle of an equilateral triangle determine each other.
    1. In a right triangle with standard notation and right angle at C, show that .
    2. Deduce that in an isosceles triangle with base b and side a, summit at B, and one base angle B, and one base angle at A,

      (HINT: Drop the altitude to the base.)

  3. In a right triangle with right angle at C (and standard notation), show that

    where .

  4. Let s denote the semiperimeter of . Show that

  5. Let h denote the length of the altitude from vertex B. Show that

    ,

    where S is the constant ratio in the Hyperbolic Law of Sines. Let . What is the significance of H? It is the hyperbolic analogue of what Euclidean quantity? Show that .

  6. Show that . This is the Hyperbolic Heron's Formula.
  7. An equilateral triangle in the hyperbolic plane has side 1.4 units. Find the angles, the hyperbolic area and H.