Method of Coordinates II (en)
Sergey Sobolev
1. Draw the points A(4,3), B(1,7), C(-3,4), and D(0,0). If you
have drawn them correctly, you have the vertices of a square.
What is the length of the sides of this square? What is its
area? Find the coordinates of the midpoints of the sides of the
square. Can you show that ABCD is a square?
2. In a plane the points A(0,0), B(3,1), and D(-1,2) are given.
What coordinates must the point C have so that the quadrangle
ABCD will be a parallelogram?
3. Try to decide by yourself which sets of points are defined
by these relations:
(a) |x|=|y|;
(b) x/|x|=y/|y|;
(c) |x|+x=|y|+y;
(d) |x|+|y|=1;
(e) |x|-|y|=1;
(f) |x+y|+|x-y|=2;
(g) x^2-y^2\ge 0;
(h) xy\ge 1.
4. The points A(x_1,y_1) and B(x_2,y_2) are adjacent vertices
of a parallelogram ABCD with center at O(0,0). What are the
coordinates of points C and D?
5. In a plane the points A(0,0), B(x_1,y_1), and D(x_2,y_2) are
given. What coordinates must the point C have so that the
quadrangle ABCD will be a parallelogram?
6. Apply the formula for the distance between two points to
prove the well-known theorem: In a parallelogram the sum of the
squares of the sides is equal to the sum of the squares of the
diagonals. [Hint. See problem 4 or problem 5.]
7. Using the method of coordinates, prove the following
theorem: if ABCD is a rectangle, then for an arbitrary point M
the equality AM^2+CM^2=BM^2+DM^2 is valid. What is the most
convenient way of placing the coordinate axes?
8. What set of points is specified by the equation
x^2+y^2\le 6x+8y?
9. Find the locus of points M the difference of the squares of
whose distances from two given points A and B is equal to a
given value c.
10. Find the locus of points M the sum of the squares of whose
distances from the vertices of given square is equal to a given
value c. For what values of c does the problem have a solution?
-- S.Sobolev