A(2,2) B(11,4) C(14,8): AC is the perpendicular bisector of BD. Find the coordinates of D

Answer 1

Line segment AC is part of a line. Use the two-point formula to find the equation of the line:

(y-2) = ( (8-2)/(14-2) ) (x-2)

y = (1/2)x + 1 [note: slope = 1/2]

AC is perpendicular to the line of segment BD. The line containing segment BD has a slope of (-2), the negative reciprocal of (1/2). Use the point-slope formula to determine the equation of that line:

(y-4) = (-2)(x-11)

y = -2x + 26 [equation of line with segment BD]

To find the intersection point of line segment BD and AC. This is the midpoint M of segment BD.

(1/2)x + 1 = -2x + 26

x + 2 = -4x + 52

5x = 50

x = 10

then, use either equation to find y

y = (1/2)(10) + 1

y = 6

The line segments intersect at point M(10.6). This is the midpoint ("bisector") of segment BD.

Let the coordinates of D be (a,b) and use the midpoint formula:

10 = (11+a) / 2

20 = 11 + a

a = 9

6 = (4+b) / 2

12 = 4 + b

b = 8

Point D is (9,8).

Answer 2

The point D is (9,8) It helps to understand the following better if you graph the points and lines

AC rises 6 as it runs 12. It's slope = m = 6/12 =1/2 The bisector BD has a slope = -2, the negative inverse of AC's slope. (4-y)/ (11 - x)=-2 where (x,y) gives the coordinates of point D

cross multiply: 4-y=22-2x or y=-2x+26 is the equation of BD

the equation of AC is (y-2)/(x-2) = 1/2

cross multiply to get 2y-4 = x-2

add 4 to both sides : 2y = x+2

divide by 2 y = (1/2)x+1

set the equations equal to solve for the intersection point (10,6)

y=(1/2)x+1 = y = -2x + 26

(1/2)x + 1 = -2x +26

(5/2)x = 25

5x = 50

x = 10

y = -2(10) + 26 = -20+26 = 6

the intersection point is (10,6)

Point D is the other side of the line AC, the same distance from (10,6) as B (11,4) is

(10,6) is the midpoint between B and D, the average of their coordinates

10 to 11 = 1 increase, so decrease 10 one to get 9 as the x coordinate of D

6 to 4 = 2 decrease, so increase 6 by 2 to get 8

D is (9,8)

It's a bisector which means AC intersects BD at BD's midpoint. The midpoint's x coordinate is half the x coordinates of B and D: (1/2) x (11+x)..

AC is a long line, BD is relatively short perpendicular line

AC is a "bisector" of BD. Let I= the intersection point of the 2 lines, (10,6)

The distance BI = ID (9,8) to (10,6) is the same distance as from (10,6) to (11,4), the square root of (11-10)²+(6-4)² = square root of 5, about 2.2