Question

Do the point (3,2),(-2,-3) and (2,3) from a triangle ? If so, name the type of triangle formed.

GIVEN:

Three POINTS

(3,2) , (-2,-3) , (2,3)

TO FIND:

Do these points form a triangle?, If yes then which type of triangle is formed?

SOLUTION:

There are two ways of finding while these points are forming triangle or not.

By PLOTTING the points in a graph

By calculating the area of figure formed by these points.

So, here we will do it by finding the area

Using FORMULA to calculate area of triangle

→ Area = 1/2 \mid∣ [ x_1x

1

( y_2y

2

- y_3y

3

) + x_2x

2

( y_3y

3

- y_1y

1

) + x_3x

3

( y_1y

1

- y_2y

2

) ] \mid∣

→ Area = 1/2 | [ 3 ( -3 - (3) ) + (-2) ( 3 - 2 ) + 2 ( 2 - (-3) ) ] |

→ Area = 1/2 | [ -18 - 2 + 10 ] |

→ Area = 1/2 | [ -10 ] |

→ Area = 5 sq. units

Since, the area of figure formed by three points is not zero, it means three points are forming a triangle.

Now,

Let us take

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

Then, Finding distances using distance formula

Distance = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]

→ AB = √[ (3 - (-2))² + ( 2 - (-3))² ]

→ AB = √( 25 + 25 ) = √50 units

→ BC = √[ (-2 - 2)² + (-3 - 3)² ]

→ BC = √( 16 + 36 ) = √52 units

→ AC = √[ (3 - 2)² + (2 - 3)² ]

→ AC = √( 1 + 1 ) = √2 units

Here, by Considering the lengths of sides of triangle

we can conclude that, lengths of AB, BC, and AC are forming a Pythagorean triplet;

BC² = AB² + AC²

( √52 )² = ( √50 )² + ( √2 )²

It means these lines are forming a right angled triangle.