B (-11, 0), D (13, 12), and E (1, 6).
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
Then solve for
Plugging this back in (1), we get:
So the equation of the line (AC) is
With this, we can find the coordinates of point E,
because it is the middle of the segment [AC].
Thus, point E has coordinates (1, 6).
Now, we want the equation of the line (BD), again in the form
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,
But remember that this line also passes through E, so we have:
and solve for
So the equation of line (BD) is:
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 (
Applying this to the equation of line (BD):
and solve for
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.
multiplied by 2 is 24. So,
multiplied by 2 is 12. So,
Point D has coordinates (13, 12).