# Question #9f8d5

##### 1 Answer

#### 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

(1):

and

(2):

Then solve for

which becomes:

i.e.

Plugging this back in (1), we get:

i.e.

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].

Similarly,

**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,

Similarly,

multiplied by 2 is 12. So,

**Point D has coordinates (13, 12).**