HYPERBOLIC GEOMETRY #7 Find the area of the super triangle with vertices.... (Th

Question

HYPERBOLIC GEOMETRY #7
Find the area of the super triangle with vertices.... (There is typo as the word "triangle" is written twice) PLEASE JUSTIFY EACH STEP Please don't respond unless you're sure.
very Lambert quadrilateral the following properties: (b) Its fourth has It a parallelogram. is (c) It is a quadrilateral 3. fine the defect of a triangle AABC by If D is a point on Bc,6(AABD)-6, and AACD-a2. hat is 6(AABc)? 4. se problem 3 to prove the AAA Tho in plane: Two triangles whose correspon ing angles are equal are congruent. eorem what might be called an Angle-Angle-Angle Theorem No such This is result is true in Euclidean geometry. It also be stated as saying that there are no simi triangles in hyperbolic can triangles with same ang must be the geometry (unless they are congruent), since the same size. 5. Draw a picture of a triangle with defect 180. 6. Prove that any two super triangles with all vertices on the boundary are congruent. 7. Find the area of the super triangle triangle with vertices 1, i, and 1+ V2.

Explanation / Answer

The vertices of the triangle are 1, i and 1+ sq(2).

Let z1 = x1 + iy1 = 1, so x1 = 1 and y1 =0

That is (x1,y1) = ( 1,0)

Let z2 = x2 + iy2 = i, so x2 = 0 and y2 =1

That is (x2,y2) = (0, 1)

Let z3 = x3 + iy3 = 1+sq(2), so x3 = 1+sq(2) and y3 =0

That is (x3,y3) = ( 1+sq(2),0)

Now the area of triangle is given by,

A = Abs |1/2 Det{x1   y1   1)|
x2   y2   1
x3   y3   1

= (1/2) Det | 1 0 1|

0 1 1

(1+sq(2) 0 1

=(1/2)( 1( 1-0) -0 - 1 ( 1 + sq(2)})

=(1/2)|( -sq(2)|

= 1/sq(2)

=1.414 sq units

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