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

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Answer 1 · 2016-06-20T18:00:27

One end-pt. of all perp. bsctrs. is $O(169/34,113/34)$ & other end. pts. are $D(11/2,2),E(5,7/2), & F(5/2,5/2).$

Length of one perp.bsctr. $OD$ is $sqrt(2349)/34.$

Let us name the vertices of $Delta$ as $A(2,4),B(3,1),C(8,3)$ & let the mid-pts. of sides $BC,CA,AB$ be $D,E,F$ resp.

Clearly, the mid-pts. are $D(11/2,2),E(5,7/2), & F(5/2,5/2).$

We know that three perp. bsctrs. of sides of a $Delta$ are concurrent at a pt., known as the Circumcentre of $Delta ABC.$ Let us call it $O.$

To find $O$ , we find the eqns. of two perp.bstrs., namely, $OD & OE.$

Eqn. of $OD$ :-

$OD$ is perp. to $BC$ , & slope of $BC$ is $(3-1)/(8-3)=2/5,$ so, slope of $OD$ must be $-5/2$ . In addition, $D in OD.$

$:.$ eqn. of $OD$ is, $y-2=-5/2(x-11/2),$ or, $4y-8=-5(2x-11)=-10x+55,$ i.e., $10x+4y=63...........(1).$

On the same line, we can work out the Eqn. of $OE$ as $12x-2y=53......(2)$

Solving (1) & (2), we get $O(169/34,113/34)$

Length $OD =sqrt{(169/34-11/2)^2+(113/34-2)^2}=sqrt(2349)/34$

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