### Video Transcript

π΄π΅πΆπ· is a quadrilateral. The coordinates of the points π΄, π΅, and πΆ are one, two; negative seven, one; and zero, negative three, respectively. Find the coordinates of the point π· 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, weβre ready to add the points we do know. π΄ is one, two; π΅ is negative seven, one; and πΆ is zero, negative three. Weβll connect those three points together. And then we need to think about the properties of this rhombus.

We know that whatever point π· is, it must satisfy that all four sides are the same length and that opposite sides are parallel. This means that line segment π΄π· must be parallel to line segment π΅πΆ and that line segment πΆπ· must be parallel to the line π΄π΅. But how do we translate that into coordinates for point π·? First, letβs find the slope for line segment π΄π΅. If weβre going from point π΅ to point π΄, weβre going up one and to the right eight. And this tells us that from point πΆ to point π· must be up one, right eight. Up one is from negative three to negative two, and right eight would be from zero to eight. π· would be located at positive eight along the π₯-axis and negative two along the π¦-axis.

Letβs check and see if this is true. One way to do that is to check that line π΄π· is, in fact, parallel to line π΅πΆ. To get from point π΅ to point πΆ, we have to move down four and right seven. If weβve correctly calculated point π·, it should be four down from π΄ and seven to the right. Weβve gone from positive two along the π¦-axis to negative two, that is, down four. And weβve gone from positive one along the π₯-axis to positive eight, and that is seven places right. If we connect these points, weβll see our rhombus such that the line π΄π΅ is parallel to the line πΆπ· and the line π΅πΆ is parallel to the line π΄π·. And all four of these sides are equal in length. And so we say that point π· would be located at eight, negative two.