HOW TO CHECK WHETHER THE GIVEN POINTS FORM A RIGHT TRIANGLE

The following steps would be useful to check whether the given points form a right triangle. 

Step 1 :

Using distance formula to find the length of each side of the triangle. 

d = √[(x2 - x1)2(y2 - y1)2]

Step 2 :

Check Pythagorean Theorem for the lengths of the sides. 

That is, square of the larger side has to be equal to sum of the squares of other two sides. 

Step 3 :

Decision making :

If the lengths of the sides satisfy Pythagorean Theorem, then the points will form a right triangle, otherwise they won't.  

In each case, examine whether the points form a right triangle. 

Example 1 :

A(-3, -4), B(2, 6), C(-6, 10)

Solution :

Distance between the points A and B :

AB = √[(x2 - x1)2 + (y2 - y1)2]

Substitute (x1, y1) = (-3, -4) and (x2, y2) = (2, 6).

= √[(2 + 3)2 + (6 + 4)2]

= √[52 + 102]

= √[25 + 100]

= √125

Distance between the points B and C :

BC = √[(-6 - 2)2 + (10 - 6)2]

= √[(-8)2 + 42]

= √[64 + 16]

= √80

Distance between the points A and C :

= √[(-6 + 3)2 + (10 + 4)2]

= √[(-3)2 + 142]

= √[9 + 196]

= √205

AC2 = (√205)

= 205 ----(1)

AB2 + BC= (√125)+ (√80)2

= 125 + 80

= 205 ----(2)

From (1) and (2),

AC2 = AB+ BC2

The points A, B and C form a right triangle. 

Example 2 :

P(7, 1), Q(-4, -1), R(4, 5)

Solution :

Distance between the points P and Q :

PQ = √[(-4 - 7)2 + (-1 - 1)2]

= √[(-11)2 + (-2)2]

= √[121 + 4]

= √125

Distance between the points Q and R :

QR = √[(4 + 4)2 + (5 + 1)2]

= √[82 + 62]

= √[64 + 36]

= √100

Distance between the points P and R :

PR = √[(4 - 7)2 + (5 - 1)2]

= √[(-3)2 + 42]

= √[9 + 16]

= √25

PQ2 = (√125)

= 125 ----(1)

QR2 + PR= (√100)+ (√25)2

= 100 + 25

= 125 ----(2)

From (1) and (2),

PQ2 = QR+ PR2

The points P, Q and R form a right triangle. 

Example 3 :

P(4, 4), Q(3, 5), R(-1, -1)

Solution :

Distance between the points P and Q :

PQ = √[(3 - 4)2 + (5 - 4)2]

= √[(-1)2 + 12]

= √[1 + 1]

= √2

Distance between the points Q and R :

QR = √[(-1 - 3)2 + (-1 - 5)2]

= √[(-4)2 + (-6)2]

= √[16 + 36]

= √52

Distance between the points P and R :

PR = √[(-1 - 4)2 + (-1 - 4)2]

= √[(-5)2 + (-5)2]

= √[25 + 25]

= √50

QR2 = (√52)

= 52 ----(1)

PQ2 + PR= (√2)+ (√50)2

= 2 + 50

= 52 ----(2)

From (1) and (2),

QR2 = PQ+ PR2

The points P, Q and R form a right triangle. 

Example 4 :

A(2, 0), B(-2, 3), C(-2, -5)

Solution : 

Distance between the points A and B :

= √[(-2 - 2)2 + (3 - 0)2]

= √[(-4)2 + 32]

= √[16 + 9]

= √25

Distance between the points B and C :

BC = √[(-2 + 2)2 + (-5 - 3)2]

= √[0 + (-8)2]

= √64

Distance between the points A and C :

= √[(-2 - 2)2 + (-5 - 0)2]

= √[(-4)2 + (-5)2]

= √[16 + 25]

= √41

BC2 = (√64)

= 64 ----(1)

AB2 + AC= (√25)+ (√41)2

= 25 + 41

= 66 ----(2)

From (1) and (2),

BC2 ≠ AB+ AC2

Since Pythagorean theorem is not satisfied, the points A, B and C do not form a right triangle. 

Example 5 :

A(0, 0), B(5, 0), C(0, 6)

Solution : 

Distance between the points A and B :

= √[(5 - 0)2 + (0 - 0)2]

= √[52 + 0]

= √25

Distance between the points B and C :

BC = √[(0 - 5)2 + (6 - 0)2]

= √[(-5)2 + 62]

= √[25 + 36]

= √61

Distance between the points A and C :

A(0, 0) C(0, 6)

= √[(0 - 0)2 + (6 - 0)2]

= √[02 + 62]

= √36

BC2 = (√61)

= 61 ----(1)

AB2 + AC= (√25)+ (√36)2

= 25 + 36

= 61 ----(2)

From (1) and (2),

BC2 = AB+ AC2

The points A, B and C form a right triangle. 

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