𝐴𝐵𝐶𝐷 is a quadrilateral, in which 𝐴 is the point one, two; 𝐵 is the point one, three; 𝐶 is the point five, three; and 𝐷 is the point five, two. Determine its type using vectors. (A) Parallelogram, (B) trapezoid, (C) rectangle, or option (D) kite.
In this question, we’re given a quadrilateral 𝐴𝐵𝐶𝐷 and we’re given the coordinates of all four of its vertices. We need to determine the type of quadrilateral 𝐴𝐵𝐶𝐷 is, and we need to do this by using vectors. To do this, we’ll start by sketching points 𝐴, 𝐵, 𝐶, and 𝐷. And since we’re given the coordinates of points 𝐴, 𝐵, 𝐶, and 𝐷, we can even sketch this to scale.
In particular, we note that points 𝐴 and 𝐵 share the same 𝑥-coordinate. However, 𝐵 is one unit higher than point 𝐴. Similarly, point 𝐶 shares the same 𝑦-coordinate as point 𝐵. However, its 𝑥-coordinate is four higher. And finally, point 𝐷 is one unit below point 𝐶. This gives us a shape which looks like the following. And of course we can determine that all of the angles are right angles because we’re only moving vertically or horizontally.
At this point, it seems we’ve sketched a rectangle, which is the correct answer. However, we should check this by using vectors, since we’re told to in the question.
To do this, let’s find the vector from 𝐴 to 𝐵. We can find this vector by recalling the vector from 𝐴 to 𝐵 is equal to the position vector of 𝐁 minus the position vector of 𝐀. And the position vector of a point has components equal to the coordinates of the point. So we have the vector one, three minus the vector one, two.
And remember, we evaluate the difference between two vectors component-wise. In other words, we need to find the difference in the corresponding components. This gives us the vector one minus one, three minus two. And we can evaluate each component. We get the vector zero, one, which agrees with our diagram. This tells us to move from point 𝐴 to point 𝐵, we move one unit upwards.
We can follow the exact same process to find the vector from 𝐷 to 𝐶. The vector from 𝐷 to 𝐶 will be the position vector of 𝐂 minus the position vector of 𝐃. That’s the vector five, three minus the vector five, two. And once again we evaluate the difference between two vectors component-wise. We once again get the vector zero, one. And now we can notice something interesting. The vector from 𝐴 to 𝐵 is exactly equal to the vector from 𝐷 to 𝐶. So, not only are the vectors parallel, we can also note that they have the same length.
Let’s now follow the same process for the vector from 𝐵 to 𝐶 and the vector from 𝐴 to 𝐷. First, the vector from 𝐵 to 𝐶 is the position vector of 𝐂 minus the position vector of 𝐁. That’s the vector five, three minus the vector one, three. And once again, we evaluate this difference component-wise. We get the vector five minus one, three minus three, which simplifies to give us the vector four, zero. In other words, to move from point 𝐵 to point 𝐶, we move four units right.
And finally we can follow the same process to determine the vector from 𝐴 to 𝐷. It’s the vector five, two minus the vector one, two. We evaluate this component-wise and simplify. We get the vector four, zero. So once again we can see something interesting. The vector from 𝐵 to 𝐶 is equal to the vector from 𝐴 to 𝐷. And if the vectors are equal, this means they have the same direction. So the vectors are parallel. They also have the same magnitude, which means the vectors will have the same length.
And it is worth noting we know that side 𝐵𝐶 is four times the length of side 𝐴𝐵. So this is not a square. Therefore, by using vectors, we were able to show the quadrilateral 𝐴𝐵𝐶𝐷 is a rectangle, which is option (C).