Video: Finding the Coordinates of a Point That Would Make a Rhombus given the Coordinates of Three Other Points

π΄π΅πΆπ· is a quadrilateral. The coordinates of the points π΄, π΅, and πΆ are (1, 2), (β7, 1) and (0, β3) respectively. Find the coordinates of the point π· if the figure is a rhombus.

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Video Transcript

π΄π΅πΆπ· is a quadrilateral. The coordinates of the points π΄, π΅, and πΆ 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, weβre ready to add the points we do know. π΄ is one, two; π΅ is negative seven, one; and πΆ is zero, negative three. Weβll 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΅πΆ and that line segment πΆπ· must be parallel to the line π΄π΅. But how do we translate that into coordinates for point π·? First, letβs find the slope for line segment π΄π΅. If weβre going from point π΅ to point π΄, weβre going up one and to the right eight. And this tells us that from point πΆ 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.

Letβs 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΅πΆ. To get from point π΅ to point πΆ, we have to move down four and right seven. If weβve correctly calculated point π·, it should be four down from π΄ and seven to the right. Weβve gone from positive two along the π¦-axis to negative two, that is, down four. And weβve gone from positive one along the π₯-axis to positive eight, and that is seven places right. If we connect these points, weβll see our rhombus such that the line π΄π΅ is parallel to the line πΆπ· and the line π΅πΆ 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.