Question #9f8d5

1 Answer
Sep 21, 2015

Answer:

B (-11, 0), D (13, 12), and E (1, 6).

Explanation:

The steps:
1. find equation of line (AC)
2. find coordinates of point E.
3. find equation of line (BD) passing through E.
4. find coordinates of B and D (note, 3 and 4 goes together)

First, find the equation of the line that passes through A and C.
A is at (-1, 10)
C is at (3, 2)
So the equation of the line (in the form #y=ax+b# where #a# and #b# are the unknown) is:
(1): #10=-a+b#
and
(2): #2=3a+b#

Then solve for #b# and #a#. (1) minus (2) gives:
#10-2=-a+b-3a-b#
which becomes:
#8=-4a#
i.e. #a=-2#
Plugging this back in (1), we get:
#10=2+b#
i.e. #b=8#
So the equation of the line (AC) is #y=-2x+8#.

With this, we can find the coordinates of point E,
because it is the middle of the segment [AC].
#Delta x=x_C - x_A=3-(-1)=4# divided by two is 2. Then #x_E=x_A+(Delta x)/2 = -1 + 2=1#

Similarly,
#Delta y=y_C - y_A=2-10=-8# divided by two is -4. Then
#y_E=y_A+(Delta y)/2 = 10-4 = 6#
Thus, point E has coordinates (1, 6).

Now, we want the equation of the line (BD), again in the form #y=a'x+b'#.
We know that the two diagonals are perpendicular to each other because it is one of the properties of a rhombus (something you should know).
So we know that the product of their slopes should be -1 (that's also something you should know).
That is to say,
#a*a'=-1# but since we know #a=-2#, we have:
#a'=1/2#

But remember that this line also passes through E, so we have:
#6=1/2 *1 +b'#
and solve for #b'#:
#b'=6- 1/2 = 11/2#

So the equation of line (BD) is:
#y=1/2 x + 11/2#

We can now get the coordinates of B.
We need to use the extra piece of information given in the text.
They say "the point B lies on the x-axis".
That is to say, "the coordinates of point B is (#x_B#, 0)".
Applying this to the equation of line (BD):
#0=1/2 x_B +11/2#
and solve for #x_B#, we get:
#x_B=-11#
Point B has coordinates (-11, 0).

Now, we can get the coordinates of point D.
Since E is the midpoint of segment [BD],
D is twice as far from B than E.
#Delta x =x_E - x_B = 1-(-11)=12#
multiplied by 2 is 24. So,
#x_D= x_B + 2Delta x = -11+24=13#.

Similarly,
#Delta y = y_E - y_B = 6-0=6#
multiplied by 2 is 12. So,
#y_D= y_B +2Delta y = 0 +12=12#
Point D has coordinates (13, 12).