# Question Video: Determining the Type of a Quadrilateral by Using Vectors Mathematics

Complete the following. If π΄π΅πΆπ· is a quadrilateral and ππ = ππ and theyβre parallel, then the quadrilateral can always be classified as οΌΏ. [A] a trapezoid [B] a parallelogram [C] a kite [D] an isosceles trapezoid [E] a rectangle

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### Video Transcript

Complete the following. If π΄π΅πΆπ· is a quadrilateral and the vector from π to π is equal to the vector from π to π and theyβre parallel, then the quadrilateral can always be classified as blank. Is it option (A) a trapezoid, option (B) a parallelogram, option (C) a kite, option (D) an isosceles trapezoid, or option (E) a rectangle?

In this question, weβre given a quadrilateral π΄π΅πΆπ·, and weβre told that the vector from π to π is equal to the vector from π to π. This means they have the same magnitude and direction, so we can conclude that theyβre parallel. We need to use this information to determine what type of quadrilateral π΄π΅πΆπ· is. And to determine this, we need to note weβre only really given one piece of information: the vector from π to π is exactly equal to the vector from π to π. And although this tells us they have the same magnitude and direction, thereβs a lot of different ways we could sketch these vectors.

For example, we could sketch one vector directly above the other, as shown. We could then note that this shape represents a square. However, it doesnβt necessarily need to be this way. For example, we couldβve drawn our vectors even further apart. Then we would have had a rectangle. But this is not the only change we couldβve made. We couldβve also drawn our vectors not one above the other. We couldβve drawn them separately in the plane. And sketching the remaining sides of quadrilateral π΄π΅πΆπ· gives us a shape which looks like a parallelogram.

However, we canβt answer this question just by looking at this graphically. We need to prove that this is a parallelogram. And to do this, we need to show that the opposite sides are parallel. We already know the side from π to π is parallel to the side from π to π. So we need to show this for the other two sides.

We can do this by using vectors. We can find an expression for the vector from π to π by following the vertices of our parallelogram. We get that the vector from π to π is equal to the vector from π to π added to the vector from π to π added to the vector from π to π. This is just an application of the triangle rule for the addition of vectors.

But now we can notice something interesting. We have ππ added to vector ππ, and weβre told in the question that vector ππ is equal to vector ππ. And the vector from π to π is exactly equal to the vector from π to π. However, we switch its direction, so the vector from π to π is negative the vector from π to π. So we can rewrite this as the vector from π to π added to the vector from π to π minus the vector from π to π.

But now we can use the fact that the vector from π to π is equal to the vector from π to π. We can replace this in our expression. And then we see the vector from π to π and subtracting the vector from π to π cancel to give the zero vector. So this simplifies to give us that the vector from π to π is equal to the vector from π to π. And if the two vectors are equal, this means they have the same magnitude and direction. And we canβt refine this any further. As weβve already shown, we couldβve had a rectangle, a square, or just a general parallelogram. Therefore, we were able to show if π΄π΅πΆπ· is a quadrilateral and the vector from π to π is equal to the vector from π to π, then quadrilateral π΄π΅πΆπ· is a parallelogram.