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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