𝐴𝐵𝐶𝐷 is a quadrilateral. The coordinates of the points 𝐴, 𝐵, and 𝐶 are one, two; negative seven, one; and zero, negative three, respectively. Find the coordinates of the point 𝐷 if the figure is a rhombus.
First of all, we can think about some properties that we know about a rhombus. A rhombus is quadrilateral where all four sides are the same length. In a rhombus, the opposite sides are parallel to each other, making it a parallelogram, and its diagonals are perpendicular to each other. One strategy for solving this problem is graphing. Once we sketch our graph, we’re ready to add the points we do know. 𝐴 is one, two; 𝐵 is negative seven, one; and 𝐶 is zero, negative three. We’ll connect those three points together. And then we need to think about the properties of this rhombus.
We know that whatever point 𝐷 is, it must satisfy that all four sides are the same length and that opposite sides are parallel. This means that line segment 𝐴𝐷 must be parallel to line segment 𝐵𝐶 and that line segment 𝐶𝐷 must be parallel to the line 𝐴𝐵. But how do we translate that into coordinates for point 𝐷? First, let’s find the slope for line segment 𝐴𝐵. If we’re going from point 𝐵 to point 𝐴, we’re going up one and to the right eight. And this tells us that from point 𝐶 to point 𝐷 must be up one, right eight. Up one is from negative three to negative two, and right eight would be from zero to eight. 𝐷 would be located at positive eight along the 𝑥-axis and negative two along the 𝑦-axis.
Let’s check and see if this is true. One way to do that is to check that line 𝐴𝐷 is, in fact, parallel to line 𝐵𝐶. To get from point 𝐵 to point 𝐶, we have to move down four and right seven. If we’ve correctly calculated point 𝐷, it should be four down from 𝐴 and seven to the right. We’ve gone from positive two along the 𝑦-axis to negative two, that is, down four. And we’ve gone from positive one along the 𝑥-axis to positive eight, and that is seven places right. If we connect these points, we’ll see our rhombus such that the line 𝐴𝐵 is parallel to the line 𝐶𝐷 and the line 𝐵𝐶 is parallel to the line 𝐴𝐷. And all four of these sides are equal in length. And so we say that point 𝐷 would be located at eight, negative two.