1.2. LINES 17
Problem 1.36 Consider the points (1,2) and (3,4).
1. How many points .x; y/ are there that form the vertices of a right triangle together with
the given points?
2. Find them.
Problem 1.37 Prove or disprove that the points (1,3), (2,0), (4,4), and (5,1) are the vertices of
a square.
Problem 1.38 Construct a procedure for testing four points to see if they are the vertices of a
square and demonstrate it on the points, (0,3), (2,0), (3,5), and (5,2), that do form the vertices
of a square.
Problem 1.39 Find three points that, together with (0,0), are not the vertices of a rectangle.
Give a reason your answer is correct.
Problem 1.40 True or false, and justify your answer.
• Any three distinct points not on the same line are the vertices of a triangle.
• Any four distinct points, no three of which are on the same line, are the vertices of a
quadrilateral.
Problem 1.41 Suppose that t is time. Which of the following sets of points lie on a single line
in the plane? In some cases plotting the points may help. You may want to consult Section 2.3
if you are unfamiliar with the sine and cosine functions.
1. Points: .3t; t /
2. Points: .cos.t/; cos.t //
3. Points: .cos.t/; sin.t //
4. Points: .t; t
2
t C 1/
5. Points: .t
2
; 2t
2
C 1/
Problem 1.42 For the points, (0,0), (1,2), (2,1), (3,5), and (2,4), there are ten possible lines
that go through two of these points. What is the largest slope and the smallest slope of these
lines?
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