Question

Find the coordinates of the point equidistant from three given points A(5,1), B(-3.-7) and
C(7,-1).​

Answer 1

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Answer 2

Given : Points -

• A(5, 1)

• B(3, 7)

• C(7, 1)

To find : The coordinate of the point equidistant from those points.

Answer :

Let's assume that the coordinate is D(x, y).

Hence, DA = DB = DC.

Let's find the distance of DA first.

Formula to find the distance between two points :

Distance = $\sqrt{(x^{2} - x^{1} )^{2}+(y^{2}-y^{1})^{2} }$

From points A(5, 1) and D(x, y) , we have:

• x₁ = 5

• x₂ = x

• y₁ = 1

• y₂ = y

Substituting them into the formula,

$\bold{Distance} =\sqrt{(x-5)^{2}+(y-1)^{2}}$

                $=\sqrt{{x^{2} - 10x + 25+y^{2}-2y+1}$

               $\bold{=\sqrt{x^{2}-10x-2y+26+y^{2}}}$  

Now, let's find the distance of DB.

From points B(3, 7) and D(x, y), we have:

• x₁ = 3

• x₂ = x

• y₁ = 7

• y₂ = y

Substituting them into the formula,

$\bold{Distance} = \sqrt{(x-3)^{2}+ (y-7)^{2}}$

                $=\sqrt{x^{2}-6x+9+y^{2}-14y+49}\\\bold{=\sqrt{x^{2}-6x-14y+58+y^{2}}}$

Now, let's find the distance of DC.

From points C(7, 1) and D(x, y), we have:

• x₁ = 7

• x₂ = x

• y₁ = 1

• y₂ = y

Substituting them into the formula,

$\bold{Distance} = \sqrt{(x-7)^{2}+(y-1)^{2}}$

                $=\sqrt{x^{2} - 14x + 49 + y^{2}-2y +21} \\\bold{= \sqrt{x^{2}-14x-2y +50+y^{2}}}$

Now, let's equate all the distances to each other.

$=\sqrt{x^{2}-10x-2y+26+y^{2}}$ = $=\sqrt{x^{2}-6x-14y+58+y^{2}}$  = $= \sqrt{x^{2}-14x-2y +50+y^{2}}$

Squaring all the roots,

-10x-2y+26 = -6x-14y+58 = -14x-2y +50

Let's consider the first two parts.

-10x-2y+26 = -6x-14y+58

⇒ -10x + 6y -2y + 14y = 58 - 26

⇒ - 4x + 12y = 32

⇒ -x + 3y = 8  → [Equation (i)]

Now, let's consider the first and the last parts.

-10x-2y+26 = -14x-2y +50

⇒ -10x + 14 x = 50 - 26

⇒  4x = 32

⇒ x = 8

Now we know that the value of x, let's substitute it into Equation 1.

-8 + 3y = 8

⇒ 3y = 0

⇒ y = 0

∴  The coordinate of the point which is equidistant from points A(5, 1), B(3, 7) and C(7, 1) is D(8, 0).