On the coordinate plane, the segment from Q(6,–5) to R(–4,–5) forms one side of a rectangle. The rectangle has an area of 140 square units. Select all of the points that could be a vertex of the rectangle.
Question
On the coordinate plane, the segment from Q(6,–5) to R(–4,–5) forms one side of a rectangle. The rectangle has an area of 140 square units. Select all of the points that could be a vertex of the rectangle.
Solution
The length of the segment QR can be found using the distance formula, which in this case simplifies to the absolute difference of the x-coordinates (since the y-coordinates are the same):
Length of QR = |6 - (-4)| = 10 units
The area of a rectangle is given by the formula length * width. We know the length is 10 units and the area is 140 square units, so we can solve for the width:
Width = Area / Length = 140 / 10 = 14 units
The width of the rectangle is the vertical distance, which is the difference in the y-coordinates of the vertices. Since Q and R have the same y-coordinate of -5, the other two vertices of the rectangle must have y-coordinates of -5 ± 14 = 9 and -19.
Therefore, the possible vertices of the rectangle are (6,9), (-4,9), (6,-19), and (-4,-19).
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