Find the circumcenter of the triangle ABC. A(4,3), B(6,-1), C(-2,-4). Please help I need to study and I don't know this.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should know that you may find the circumcenter of triangle using the perpendicular bisectors of triangle. The point of intersection of perpendicular bisectors represents the circumcenter of triangle.

Hence, you need to find the equations of two bisectors, at least.

You should know that perpendicular bisector of a side of triangle is a line that is perpendicular to the side and it cuts the side in two equal segments.

Since the problem provides informations that help you to find the slope and the midpoint, you need to use the point slope form of equation of bisector line such that: `y - y_M_1 = m_1*(x - x_M_1)` .

`M_1`  represents the midpoint of AB

You need to know the equation that relates the slopes of two perpendicular lines  such that:

`m_(AB)*m_1 = -1`

You may find `m_(AB)`  using the following formula such that:

`m_(AB) = (y_B - y_A)/(x_B - x_A)`

`m_(AB) = (-1-3)/(6-4) => m_(AB) = -4/2 = -2`

Hence, evaluating the slope of bisector line yields:

`m_1 = -1/(-2) = 1/2`

You need to find the midpoint of AB using the following formula such that:

`x_M_1 = (x_A + x_B)/2 => x_M_1 = (4+6)/2 = 5`

`y_M_1 = (y_A + y_B)/2 => y_M_1 = (3-1)/2 = 1`

`M_1(5,1)`

Hence, you may write the equation of bisector line such that:

`y - y_M_1 = m_1*(x - x_M_1) => y - 1 = (1/2)(x - 5)`

`y = x/2 - 5/2 + 1 => y = x/2 - 3/2`

You need to find the equation of bisector line, perpendicular to BC, such that:

`y - y_M_2 = m_2*(x - x_M_2)`

You need to find the slope `m_2`  such that:

`m_2 = -1/(m_(BC))`

You need to find `m_(BC)`  such that:

`m_(BC) = (y_C - y_B)/(x_C - x_B) => m_(BC) = (-4+1)/(-2 - 6) `

`m_(BC) = 3/8`

You need to find the midpoint of BC using the following formula such that:

`x_M_2 = (x_B+x_C)/2 => x_M_2= (6-2)/2 = 2`

`y_M_2 = (y_B+y_C)/2 => y_M_2 = (-1-4)/2 = -5/2`

Hence, you may write the equation of bisector line such that:

`y + 5/2= (3/8)*(x - 2) => y = 3x/8 - 3/4 - 5/2`

`y = 3x/8 - 13/4`

You need to find the point of intersection between the bisector lines such that:

`{(y = x/2 - 3/2),(3x/8 - 13/4):} => x/2 - 3/2 = 3x/8 - 13/4`

Isolating the terms that contain x to the left side yields:

`x/2 - 3x/8 = 3/2 - 13/4`

`4x - 3x = 12 - 26 => x = -14`

`y = -7- 3/2 => y = -17/2`

Hence, evaluating the position of circumcenter yields `C(-14,-17/2).`

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