Where must the coordinates of points 𝐶 be so that 𝐴𝐵𝐶𝐷 is a parallelogram? In that case, what is the area of the parallelogram?
In this question, we’re asked to make a parallelogram. We can remember that a parallelogram has two pairs of parallel and congruent sides. So if we look at our length 𝐴𝐵, we can see that it must be five units long. Therefore, to draw a line from 𝐷 to 𝐶, this must also be a horizontal line of length five units. We can confirm the coordinates of point 𝐶 by checking that the line 𝐵𝐶 is parallel to the line 𝐴𝐷. We can see that on 𝐴𝐷, we moved three squares horizontally and seven squares vertically.
And to go from 𝐵 to 𝐶, we can see that it’s also three squares horizontally and seven squares vertically. So line 𝐵𝐶 is horizontal to 𝐴𝐷 and confirms our point 𝐶 is that the coordinate six, five. It’s worth noting at this point that our coordinate at negative four, five would also have created a parallelogram. However, we were told that our parallelogram was called 𝐴𝐵𝐶𝐷. And in order to obey naming conventions, we list the vertices of a shape in the order of travel. If 𝐶 was to the left of 𝐷, then the shape would have been called 𝐴𝐵𝐷𝐶, which is not what we were told. And therefore, 𝐶 is not at negative four, five.
And now, for the second part of the question, using our 𝐶 coordinate, we’re asked for the area of the parallelogram. We can find the area of a parallelogram by multiplying the base times the vertical or perpendicular height. So using our formula then of base times height, we have our base of five units and a height of seven. So five times seven is 35.
So our final answer then is point 𝐶 is at coordinate six, five and the area is 35.