A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Much of this solution uses the work of
Andrew I and Elizabeth F. Well
For the square with opposite vertices of co-ordinates (5, 3) and
(5, 7) respectively, you can obtain the other vertices (7, 5) and
(3, 5) by creating 4 right-angled congruent triangles. The
resulting figure is a square because: it has all its sides
congruent, and all its angles right angles.
For the opposite corners of co-ordinates (5, 3) and (6, 6),
using the same method, you get (4, 5) and (7, 4).
Veryfying that the four point sets from the problem are
A. (8, 3); (7, 8); (2, 7); (3, 2) - Yes
B. (3, 3); (7, 4); (8, 8); (4; 7) - No
C. (16, 19); (18; 22); (21, 20); (19, 17) - Yes
D. (4, 20); (21, 19); (20, 2); (3, 3) - Yes
For the general case, using the drawing from the figure.
Let the co-ordinates of one corner be (a, b), and of the
opposite be (c, d). Note that (x1, y1) and (x2, y2) are the
co-ordinates of the other two corners. The geometrical problem has
unique solution, that is obtained in the following manner:
Using the values m and p
(v) + (vi) gives
c + d = a + b + 2p
p = (c + d - a - b)/2
(v) - (vi) gives
c - d = a - b - 2m
m = (a - b + d - c)/2
Substituting these values for m and p iinto (i), (ii), (iii) and
x1 = (a + c + d - b)/2
y1 = (a + b + d - c)/2
x2 = (a + b + c - d)/2
y2 = (b + c + d - a)/2