Given triangle ABC, A(-5,2), B(7,4), C(3,-9). If D(1,3), determine if segment CD is an altitude, median, perpe

Question

Given triangle ABC, A(-5,2), B(7,4), C(3,-9). If D(1,3), determine if segment CD is an altitude, median, perpendicular bisector, or angle bisector. Show work to support your answer.

Answer 1

slope of ab = (4-2)/(7--5)=1/6
slope of cd = (3--9)/(1-3)=-6
so ab is perpendicular to cd
length of ad = sqrt((3-2)^2 + (1--5)^2)=sqrt(37)
length of db = sqrt((3-4)^2+(1-7)^2)=sqrt(37)
so CD is a perpendicular bisector on AB

Answer 2

Joinn C(3,-9) and D(1,3).

Slope of CD = (3+9)/(1-3) = -12/2 = -6

slope of AB = (4-2)/ (7+5) = 2/12 = 1/6

The product of slopes of lines AB and CD = -6*(1/6) = -1

so AB and CD are perpendicular and CD is altitude.

mid point of AB = (-5+7)/2 , (2+4)/2 = (1,3)

since coordinates of mid point AB and point D are same. D is the mid point of AB. CD is median also.

So triangles CDA and CDB are congruent and therefore

angle ACD = angle BCD.

So CD is altitude, median and perpendicular bisector of AB and angle bisector of angle ACB.