Question

32. Parallelogram ABCD has vertices A(0,0), B(2,4), and C(10,4). Find the coordinates of D.

Answer 1

Answer:

The fourth vertex is D(8, 0)

Step-by-step explanation:

Let A(0, 0), B(2, 4), and C(10, 4) be the three vertices of a parallelogram ABCD and let its fourth vertex be D(a, b).

Join AC and BD. Let AC and BD intersect at point O.

• We have known that the diagonals of a parallelogram bisect each other.  

So, O would be the midpoint of AC as well as that of BD.

$A\left(0,\:0\right)=\left(x_1,\:x_2\:\right)$

$C\left(10,\:4\right)=\left(x_2,\:x_2\:\right)$

The midpoint of AC is:

$\mathrm{Midpoint\:of\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \left(\frac{x_2+x_1}{2},\:\:\frac{y_2+y_1}{2}\right)$

$\left(x_1,\:y_1\right)=\left(0,\:0\right),\:\left(x_2,\:y_2\right)=\left(10,\:4\right)$

$=\left(\frac{10+0}{2},\:\frac{4+0}{2}\right)$

$=\left(5,\:2\right)$

The midpoint of BD is:

$=\left(\frac{a+2}{2},\:\frac{b+4}{2}\right)$

so

∵  $\frac{a+2}{2}=5$

$a+2=10$

$a=8$

∵ $\frac{b+4}{2}=2$

$b+4=4$

$b=0$

Hence, the fourth vertex is D(8, 0)