Finding coordinates of 4th point in a quadilateral

Question

A(1,5), B(4,0), C(-3,-5) are three vertices of a parallelogram ABCD.

Find the coordinates of D, the fourth vertex of the parallelogram

*What I've Done:*I've created the structure on a number plane. I was wondering do you have to find the intersection of AC (These are the points that join up to D)

Thanks!!

Have you perchance heard of the "parallelogram rule" for adding vectors? You can use it in reverse here. — hmakholm left over Monica, Oct 18 '13 at 22:14
The question is phrased as if there is only one correct answer, but this may not be the case. E.g., if the three given points were $(0,0),(1,0),(0,1)$ then the 4th point could be $(1,1)$ or $(-1,1)$ or $(1,-1)$. — Gerry Myerson, Oct 18 '13 at 22:24

score 2 · Answer 1 · answered Oct 18 '13 at 22:15

2

Hint

Write $\vec{AB}=\vec{DC}$ and solve two equations knowing that $\vec{MN}=(x_N-x_M,y_N-y_M)$.

answered Oct 18 '13 at 22:15

What do those arrows mean? – Mark Zakhem Oct 18 '13 at 22:21
$\vec{AB}$ is the vector $AB$, do you know what's a vector? – Oct 18 '13 at 22:23
No I do not! Is that like a vertice? – Mark Zakhem Oct 18 '13 at 22:25
Sami, how do you know AB and CD aren't the diagonals of the parallelogram? – Gerry Myerson Oct 18 '13 at 22:26
We've drawn the structure on a number plane – Mark Zakhem Oct 18 '13 at 22:27
Usually from the order of the letters $ABCD$ we understand that $AC$ and $BD$ are the two diagonals @GerryMyerson – Oct 18 '13 at 22:30
That may be the intended interpretation, but there's nothing in the statement of the question explicitly supporting that interpretation. – Gerry Myerson Oct 18 '13 at 22:32
@GerryMyerson 'Nothing in the statement....'? Take any geometric book and if we find: Let ABCD a parallelogram we understand without any doubt that $AC$ and $BD$ are their diagonals. – Oct 18 '13 at 22:38
A + makes your day a bit smoother. After having a hard work time. Teaching and solving maths is great for us. :-) – Mikasa Oct 20 '13 at 14:12

score 0 · Answer 2 · answered Oct 18 '13 at 22:24

0

Hint

Distance between two points $P(p_1,p_2)$ and $Q(q_1,q_2)$ in plane is $\sqrt{(q_1-p_1)^2+(q_2-p_2)^2}$

The distance between $D(x,y)$ and $C(-3,-5)$ is equal to the distance between $A(1,5)$ and $B(4,0)$. You can find $x,y$ by the above formula.

answered Oct 18 '13 at 22:24

Ömer

2,038

You are assuming $C$ and $D$ are adjacent. – Gerry Myerson Oct 18 '13 at 22:25
@GerryMyerson, yes of course. – Ömer Oct 18 '13 at 22:33
So, that means you are finding only one of the several correct answers. And, anyway, the problem doesn't need distances or square roots. – Gerry Myerson Oct 18 '13 at 22:34
I am finding $D$ from the order of letters. – Ömer Oct 18 '13 at 22:44
Fair enough. Still, I think introducing distance overcomplicates the problem. – Gerry Myerson Oct 18 '13 at 22:46
May be the problem does not need the things you mentined. But this is an alternative way not the only way. Let the Original poster select his answer from the answers. – Ömer Oct 18 '13 at 22:53

score 0 · Answer 3 · answered Oct 18 '13 at 22:31

To get from A to B, you have to add $(3,-5)$; that is, $$(1,5)+(3,-5)=(4,0)$$ If you add $(3,-5)$ to C, you get one correct value for D.

You get other correct answers if you see what you have to add to A to get to C, and then add that to B, or if you see what you have to add to B to get to C, and then add that to A.

Finding coordinates of 4th point in a quadilateral

3 Answers3