Question #9f8d5

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).