Consider the two diagrams below. Choose ONE and fill in all necessary details to prove the Pythagorean Theorem using the diagram. Make sure that each step is fully justified. (In each picture, you may...

A B
C
A B
C
Instructions: Complete the following 5 problems. For each problem (except #2), you have a choice of
two options. Please be clear about which question option you are answering.
Remember to show all necessary work and justify your answers fully.
1. (10 points) Consider the two diagrams below. Choose ONE and fill in all necessary details to
prove the Pythagorean Theorem using the diagram. Make sure that each step is fully justified.
(In each picture, you may take as given that triangle ABC is a right triangle and that the
quadrilaterals are squares).
Solution – Using Figure B
a. b.
a
a
b
a
b-a
c
With assumptions, the figure becomes
Triangle PTS and Triangle ABC are congruent by SAS
Hence, side BC = c
Now, the problem is very simple. We need to prove c (sq) = a (sq) + b(sq)
For that, we rotate the the triangles 90 degree each around its top vertex. The top one is rotated
counterclockwise. The bottom one is rotated clockwise. Something like this:-
a
a
b
b
c
P
Q
S
R
T
b
b
a
a
So it will result into this figure:-
The resulting figure is a square with side c and area c(sq)
The area of this is equal to square with sides „a‟ and „b‟
So c (sq) = a(sq) + b(sq)
This is a very famous proof by one of the top scholars of the world. Guess who 
2. (15 points) Problem-solving in Taxicab Geometry. Over the past few weeks, we have looked at
how different Euclidean features change in taxicab geometry. For example, we looked at circles
and at the midset of two points. For this problem, use problem-solving strategies to discover,
specify, and report on an additional aspect of taxicab geometry. Choose either problem a or b.
a. We have considered the midset of two points (the set of points which are equidistant from
two specified points in the plane). What can you say about the midset of two lines?
In other words, what is the locus of points (or set of points specified by) equidistant from
two different lines? You should report on your findings for both the Taxicab and
Euclidean planes. Be as precise as possible, accounting for any different possibilities.
b. We have considered the different shape of a circle in a Euclidean and Taxicab plane.
What can we say about the shape of other conic sections (e.g. Ellipse,
Parabola, or Hyperbola) in the Taxicab plane? Choose one other conic section
(Ellipse, Parabola, or Hyperbola) and report on your findings. Be as precise as possible,
accounting for any different possibilities.
(Note: the locus definitions of these conics may help you. A parabola is a locus of points
equidistant from a single point, called the focus of the parabola, and a line, called the
directrix of the parabola. Given two points, F1 and F2 (the foci), an ellipse is the locus of
points P such that the sum of the distances from P to F1 and to F2 is a constant. A
hyperbola is the locus of points P such that the absolute value of the difference between
the distances from P to F1 and to F2 is a constant.)
Solution – for problem „b‟(chosen)
Definition
The definition of a Parabola (Eucledian plane) is the locus of points equidistant from a line and a
fixed point.
Let l be a line and P a point.
Three cases are presented. Taxi cab distance is the sum along a horizontal plus a vertical. If the
point Q is on the horizontal with P then h would be the taxicab distance from P to Q. If the point
Q is on the vertical with P then v is the taxicab distance from P to Q. The TC distance from P to
the line would be the minimum for all points Q on the line.
Thus
if the slope of l is more than 1, the h is the minimum distance.
if the slope of l is less than 1, the v is the minimum distance.
if the slope of l is equal to 1, the h = v and all points between BC have the same TC length.
Similar figures could be examine for negative slopes.
The definition of a Parabola is the locus of points such that a point on the Parabola is equidistant
from a line called the directrix and a point called the focus. In Euclidean Geometry, the distance
of a point from the line is taken along the perpendicular from a point on the directrix. It makes no
difference what the slope of the line is.
Case1. |m| > 1.
|If the line is y = mx + c and we have point F(a,b), then we need to set the two TC distances equal to one
another. |x - a| + |y - b| gives us the distance from F. The distance to the line would be along a horizontal
from any point (x,y) on the locus. Therefore the coordinates of a point Q that is minimum distance from
(x,y) would be ( (y-c)/m, y). Thus the distance to the line would be |x - (y-c)/m| + |y - y| = |x - (y-
c)/m|.Thus the equation for the TC Parabola is
|x - a| + |y - b| = |x - (y-c)/m|
Case2. |m| < 1
As in case one, we have y = mx + c and point F(a,b). The distance to the line would be along the vertical
from any point (x,y) on the locus. Therefore the coordinates of a point Q that is minimum distance from
(x,y)...