SHOW THAT THE POINTS ARE THE VERTICES OF A RIGHT TRIANGLE
The following steps would be useful to check if four points form a rectangle.
Step 1 :
Find the lengths of all three sides of the triangle using distance formula.
Step 3 :
Using the lengths found, check whether Pythagorean Theorem is satisfied. That is, square of one of the sides is equal to sum of the squares of other two sides.
Example 1 :
Show that the following are the vertices of a right angled triangle.
A(-3, -4), B(2, 6), C(-6, 10)
Solution :
Distance between A and B :
Formula to find the distance between two points :
d = √[(x2 - x1)2 + (y2 - y1)2]
Substitute (x1, y1) = A(-3, -4) and (x2, y2) = B(2, 6).
= √[(2 + 3)2 + (6 + 4)2]
= √[52 + 102]
= √[25 + 100]
AB = √125
AB2 = 125
Distance between D and C :
= √[(x2 - x1)2 + (y2 - y1)2]
Substitute (x1, y1) = B(2, 6) and (x2, y2) = C(-6, 10).
= √[(-6 - 2)2 + (10 - 6)2]
= √[(-8)2 + 42]
= √[64 + 16]
DC = √80
DC2 = 80
Distance between A and C :
= √[(x2 - x1)2 + (y2 - y1)2]
Substitute (x1, y1) = A(-3, -4) and (x2, y2) = C(-6, 10).
= √[(-6 + 3)2 + (10 + 4)2]
= √[(-3)2 + 142]
= √[9 + 196]
AC = √205
AC2 = 205
From the above workings,
125 + 80 = 205
AB2 + DC2 = AC2
The side lengths AB, BC and AC satisfy Pythagorean Theorem.
So, the given points form a right triangle.
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