# A triangle has corners at (3,7), (4,1), and (8,2). What are the endpoints and lengths of the triangle's perpendicular bisectors?

Jan 26, 2018
1. End points of the perpendiclar bisectors

$D \left(6 , \frac{3}{2}\right) , E \left(\frac{11}{2} , \frac{9}{2}\right) , F \left(\frac{7}{2} , 4\right)$ with circum center $P \left(\frac{53}{10} , \frac{43}{10}\right)$

1. Lengths of the perpendicular bisectors are

$P D = \textcolor{b l u e}{2.8862} , P E = \textcolor{b l u e}{0.2828} , P F = \textcolor{b l u e}{1.8248}$

#### Explanation: A, B, C are the vertices and Let D, E, F be the mid points of the sides a, b, c respectively.

Midpoint of BC = D = (x2 + x1)/2, (y2 + y1)/2 = (4+8)/2, (1+2)/2 =
(6,3/2)

Slope of BC ${m}_{B C} = \frac{y 2 - y 1}{x 2 - x 1} = \frac{2 - 1}{8 - 4} = \frac{1}{4}$

Slope of the perpendicular bisector through D = $- \frac{1}{m} _ \left(B C\right) = - 4$

Equation of perpendicular bisector passing through mid point D using standard form of equation $y - y 1 = m \cdot \left(x - x 1\right)$

$y - \left(\frac{3}{2}\right) = - 4 \cdot \left(x - 6\right)$

$\textcolor{red}{2 y + 8 x = 51}$ Eqn (D)

Similarly,

Mid point of CA = $E \left(\frac{11}{2} , \frac{9}{2}\right)$

Slope of CA = m_(CA) = (2-7)/(8-3) = -1

Slope of the perpendicular bisector through E = $- \frac{1}{m} _ \left(C A\right) = 1$

Equation of perpendicular bisector passing through mid point E is

y-(9/2) = 1 * (x - (11/2)

$\textcolor{red}{y - x = - 1}$ Eqn (E)

Similarly,

Mid point of AB = $F \left(\frac{7}{2} , 4\right)$

Slope of AB= m_(AB) = (1-7)/(4-3) = -6

Slope of the perpendicular bisector through F = $- \frac{1}{m} _ \left(A B\right) = \frac{1}{6}$

Equation of perpendicular bisector passing through mid point F is

y - 4 = (1/6) * (x - (7/2)

$12 y - 48 = 2 x - 7$

$\textcolor{red}{12 y - 2 x = 41}$ Eqn (F)

Solving Eqns (D), (E), we get the coordinates of circumcenter P.

$\textcolor{g r e e n}{P \left(\frac{53}{10} , \frac{43}{10}\right)}$

This can be verified by solving Eqns (E), (F).
and the answer is color(green)(P (53/10, 43/10)#

Length of the perpendicular bisectors PD

$P D = \sqrt{{\left(6 - \left(\frac{53}{10}\right)\right)}^{2} + {\left(\left(\frac{3}{2}\right) - \left(\frac{43}{10}\right)\right)}^{2}} = \textcolor{b l u e}{2.8862}$

Length of perpendicular bisector PE

$E P = \sqrt{{\left(\left(\frac{11}{2}\right) - \left(\frac{53}{10}\right)\right)}^{2} + {\left(\left(\frac{9}{2}\right) - \left(\frac{43}{10}\right)\right)}^{2}} = \textcolor{b l u e}{0.2828}$

Length of perpendicular bisector PF
7/2, 4
$P F = \sqrt{{\left(\left(\frac{7}{2}\right) - \left(\frac{53}{10}\right)\right)}^{2} + {\left(4 - \left(\frac{43}{10}\right)\right)}^{2}} = \textcolor{b l u e}{1.8248}$