Question

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

Answer 1

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.

1. By plotting the points in a graph
2. 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 $\mid$ [ $x_1$ ( $y_2$ - $y_3$ ) + $x_2$ ( $y_3$ - $y_1$ ) + $x_3$ ( $y_1$ - $y_2$ ) ] $\mid$

→ 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.

Answer 2

Let the given points be,
⇒A(3,2),B(−2,−3) and C(2,3)

Therefore,
AB=
(3+2)
2
+(2+3)
2

​
=
50
​
=5
2
​
units
BC=
(−2−2)
2
+(−3−3)
2

​
=
52
​
=2
13
​
units
AC=
(3−2)
2
+(2−3)
2

​
=
2
​
units

Now, we can see that,
(2
13
​
)
2
=(5
2
​
)
2
+(
2
​
)
2

BC
2
=AB
2
+AC
2