Thread: What are the coordinates of A

A trapezoid ABCD has an area of 24 sq. units. There of its coordinates are
B(1,8) , C(2,3), and D(-6,3) . Find the coordinates of A.

can anybody help me on this problem above please.

Originally Posted by rcs

A trapezoid ABCD has an area of 24 sq. units. There of its coordinates are
B(1,8) , C(2,3), and D(-6,3) . Find the coordinates of A.

can anybody help me on this problem above please.

1. This problem has no unique solution, because with the given conditions you can draw 6 different trapezoids.

2. See attachment: There are 2 trapezoids in red, 2 trapezoids in blue, 2 trapezoids in green.

3. Take the red trapezoids (that's the simplest case!):

Point A must be situated on a parallel to the x-axis with the equation y = 8.

The area of a trapezoid is calculated by : $\displaystyle A_t = \frac{|\overline{AB}|+|\overline{CD}|}{2} \cdot h$

$\displaystyle |\overline{CD}| = 8$, $\displaystyle h = 8$

So you have to solve for $\displaystyle |\overline{AB}|$:

$\displaystyle 24 = \frac{|\overline{AB}|+8}{2} \cdot 5$

which yields $\displaystyle |\overline{AB}|= 1.6$

So you get 2 different points A: $\displaystyle A_1(-0.6, 8)$ or $\displaystyle A_2(2.6, 8)$

4. The next 4 trapezoids are much more difficult to determine.

Originally Posted by earboth

1. This problem has no unique solution, because with the given conditions you can draw 6 different trapezoids.

2. See attachment: There are 2 trapezoids in red, 2 trapezoids in blue, 2 trapezoids in green.

3. Take the red trapezoids (that's the simplest case!):

Point A must be situated on a parallel to the x-axis with the equation y = 8.

The area of a trapezoid is calculated by : $\displaystyle A_t = \frac{|\overline{AB}|+|\overline{CD}|}{2} \cdot h$

$\displaystyle |\overline{CD}| = 8$, $\displaystyle h = 8$

So you have to solve for $\displaystyle |\overline{AB}|$:

$\displaystyle 24 = \frac{|\overline{AB}|+8}{2} \cdot 5$

which yields $\displaystyle |\overline{AB}|= 1.6$

So you get 2 different points A: $\displaystyle A_1(-0.6, 8)$ or $\displaystyle A_2(2.6, 8)$

4. The next 4 trapezoids are much more difficult to determine.

From the book where I got this problem has answer but does not have a solution. It says A(-1,8). I wonder how they got it.

To begin with, I think that the answer given in your book is wrong.

I think that the wording, 'a trapazoid ABCD', implies that the points should be taken in that order in which case the point A lies between the points B and D (and outside of the triangle BCD).
With that assumption, there seem to be two possible constructions, one where AB is parallel to DC and the other where AD is parallel to CB. The first of these is the easier to deal with.

The triangle BCD has a base of length 8 and a height 5 so its area is 8*5/2=20. That means that the triangle ABD should have an area of 4.
Take AB to be the base of ABD, then its height will be 5 in which case its area will be 5*AB/2 and that has to equal 4.
From that it's easy to calculate the x coordinate of A, (and it agrees with an answer given in an earlier post).

The other possible position for A is more awkward to calculate, and since your book gives only one answer, I would guess that you are not expected to find it.

Originally Posted by rcs

From the book where I got this problem has answer but does not have a solution. It says A(-1,8). I wonder how they got it.

Same to me!

If A(-1, 8) then $\displaystyle |\overline{AB}| = 2$.

With $\displaystyle |\overline{CD}| = 8$

The area of the trapezoid is: $\displaystyle A_T = \frac{2+8}2 \cdot 5=5 \cdot 5 = 25$

But maybe this result includes the VAT

Hello, rcs!

Is there a typo in the problem?
That would explain their answer.

A trapezoid ABCD has an area of 25 sq. units.
Three of its coordinates are: B(1,8), C(2,3), and D(-6,3) .
Find the coordinates of A.

The problem is not clearly stated.
As earboth pointed out, there are six possible trapezoids.

I assume that the horizontal side $\displaystyle CD$ is the "base"
. . and $\displaystyle AB \parallel CD.$
Hence, $\displaystyle A$ is in Quadrant 2.

Code:

                    |
              (x,8) |   (1,8)
               A♥ * * * * ♥B
              *     |      *
            *       |       *
          *         |        *
        *           |         *
     D♥ * * * * * * |* * * * * ♥C
   (-6,3)           |         (2,3)
                    |
  - - - - - - - - - + - - - - - - - -
                    |

The height is $\displaystyle h = 5.$

The bases are: .$\displaystyle C\!D \,=\, 8,\;AB \,=\, 1-x$

Area of a trapezoid: .$\displaystyle A \:=\:\tfrac{h}{2}(b_1 + b_2)$

Hence, we have: .$\displaystyle \tfrac{5}{2}\big[8 + (1-x)\big] \:=\:2{\color{red}5}$

. . $\displaystyle \tfrac{5}{2}(9 -x) \:=\:25 \quad\Rightarrow\quad 9-x \:=\:10 \quad\Rightarrow\quad x \:=\:-1$

Therefore: .$\displaystyle A(\text{-}1,8)$

Yeah I gues... Thanks sir Soroban