Question

The circumcentre of the triangle formed by (2,-5), (2, 7), (4,7) is
1. (3, 1)
2. (2,-9)
3. (4, -1)
4. (3/2, 5/2)​

Answer 1

Given :  triangle formed by (2,-5), (2, 7), (4,7)

To find : circumcentre of the triangle

Solution:

Let say ( x , y ) is the  circumcenter of triangle formed by

(2,-5), (2, 7), (4,7)

Distance of circumcenter from each vertex is same

(x - 2)² + (y + 5)² = (x - 2)² + (y - 7)² = (x - 4)²  + (y - 7)²

(x - 2)² + (y + 5)² = (x - 2)² + (y - 7)²

=> (y + 5)² =  (y - 7)²

=> y² + 10y + 25 = y² -14y + 49

=> 24y = 24

=> y = 1

(x - 2)² + (y - 7)² = (x - 4)²  + (y - 7)²

=>  (x - 2)²   = (x - 4)²

=> x² -4x + 4 = x² - 8x + 16

=> 4x = 12

=> x = 3

( 3 ,1 ) the  circumcentre of the triangle formed by (2,-5), (2, 7), (4,7)

option 1 is correct

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Answer 2

Given data :

Vertices of  a triangle are (2,-5), (2,7), (4,7)

To find :

The circumcentre of the triangle.

Calculation :

Let (x,y) be the circumcentre of the triangle.

We know that, the distance from the circumcentre to any of the vertex is same.

So, using the distance between two points formula, let us equate the distances of every vertex to the circumcentre.

As in,

(x - 2)² + (y - (-5))² = (x - 2)² + (y - 7)² = (x - 4)² + (y - 7)²

(x - 2)² + (y + 5)² = (x - 2)² + (y - 7)² = (x - 4)² + (y - 7)²

Let us equate the distances of first to points to the circumference to get an equation :

(x - 2)² + (y + 5)² = (x - 2)² + (y - 7)²     (cancel (x - 2)² on both sides)

⇒ (y + 5)² = (y - 7)²

⇒ y² + 10y + 25 = y² -14y + 49           (cancel y² on both sides)

⇒ 10y + 25 = -14y + 49

⇒ 10 + 14y = 49 - 25

⇒ 24 y = 24

⇒ y = 1

The y coordinate of the circumcentre (x,y) is 1.

Finding another equation to find x coordinate :

(x - 2)²+(y - 7)² = (x - 4)²+(y - 7)²           (cancel (y - 7)² on both sides)

⇒ (x - 2)²   = (x - 4)²

⇒ x² - 4x + 4 = x² - 8x + 16                        (cancel x² on both sides)

⇒ -4x + 4 = -8x + 16

⇒ -4x + 8x = 16 - 4

⇒ 4x = 12

⇒ x = 3

The x coordinate of the circumcentre (x,y) is 3.

Therefore, (3,1) is the circumcentre of the triangle formed by (2,-5), (2, 7), (4,7).

(Option 1)

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