Question

find the circumcentre of A(2,-5),B(6,8) C(2,8)

Answer 1

Given,

Coordinates of the three vertices of ∆ABC are = A(2,-5),B(6,8) C(2,8)

To find,

The coordinate of the circumcentre of ∆ABC.

Solution,

Let, the coordinates of the circumcentre ∆ABC = Point O = x,y

Now, OA = OB = OC = Radius of the circle

Length of OA = ✓(x-2)²+(y+5)² units

Length of OB = ✓(x-6)²+(y-8)² units

Length of OC = ✓(x-2)²+(y-8)² units

Now,

OB= OC

OB² = OC²

(x-6)²+(y-8)² = (x-2)²+(y-8)²

(x-6)² = (x-2)²

x²-12x+36 = x²-4x+4

-12x+4x = 4-36

-8x = -32

x = 4

OA² = OB²

(x-2)²+(y+5)² = (x-6)²+(y-8)²

(4-2)²+(y+5)² = (4-6)²+(y-8)²

4+y²+10y+25 = 4+y²-16y+64

10y+25 = -16y+64

26y = 39

y = 1.5

Hence, the coordinate of the circumcentre is (4,1.5)

Answer 2

SOLUTION

TO DETERMINE

The circumcentre of A(2,-5),B(6,8), C(2,8)

CONCEPT TO BE IMPLEMENTED

Let ABC is a right angle triangle with ∠C as right angle and AB as hypotenuse. If we draw a circumcircle of the triangle then AB is the diameter of the circle and midpoint of AB is circumcentre of the circle

EVALUATION

Here the given points are A(2,-5) , B(6,8), C(2,8)

Now

$\sf{AB = \sqrt{ {(6 - 2)}^{2} +{ (8 + 5)}^{2} } }$

$\implies \sf{AB = \sqrt{ {4}^{2} +{13}^{2} } }$

$\therefore \: \: \sf{AB = \sqrt{185} }$

$\sf{BC = \sqrt{ {(6 - 2)}^{2} +{ (8 - 8)}^{2} } }$

$\implies \sf{BC = \sqrt{ {4}^{2} +{ 0}^{2} } }$

$\therefore \: \: \sf{BC = 4 }$

$\sf{AC = \sqrt{ {(2- 2)}^{2} +{ (8 + 5)}^{2} } }$

$\implies \sf{AC = \sqrt{ {0}^{2} +{13}^{2} } }$

$\therefore \: \: \sf{AC = 13}$

So From above

$\sf{ {AB }^{2} ={BC}^{2} + {AC }^{2} }$

∴ ABC is a right angle triangle with ∠C as right angle and AB as hypotenuse

Hence midpoint of AB is the circumcentre

Hence the circumcentre

$\displaystyle = \sf{ \bigg( \: \frac{2 + 6}{2} \:, \: \frac{ - 5+ 8}{2} \: \bigg)}$

$\displaystyle = \sf{ \bigg( \: \frac{8}{2} \:, \: \frac{ 3}{2} \: \bigg)}$

$\displaystyle = \sf{ \bigg( \: 4 \:, \: \frac{ 3}{2} \: \bigg)}$

$\displaystyle = \sf{ \bigg( \: 4 \:, \: 1.5 \: \bigg)}$

Figure : For figure refer to the attachment

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