On the coordinate plane, the segment from M(–3,12) to N(–3,5) forms one side of a rectangle. The rectangle has a perimeter of 34 units. Select all of the points that could be a vertex of the rectangle.
Question
On the coordinate plane, the segment from M(–3,12) to N(–3,5) forms one side of a rectangle. The rectangle has a perimeter of 34 units. Select all of the points that could be a vertex of the rectangle.
Solution
The length of the segment MN is the difference in the y-coordinates of M and N, which is 12 - 5 = 7 units.
Since the perimeter of the rectangle is 34 units, the sum of the lengths of all four sides of the rectangle is 34 units.
In a rectangle, opposite sides are equal in length. Therefore, the length of the other pair of opposite sides is 34 - 2*7 = 20 units. Since these two sides are equal in length, each is 20/2 = 10 units long.
The coordinates of the other two vertices of the rectangle depend on whether these sides are horizontal or vertical.
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If these sides are horizontal, they are parallel to the x-axis. Therefore, their x-coordinates change by 10 units, but their y-coordinates remain the same. The coordinates of the other two vertices are (-3+10, 12) = (7, 12) and (-3+10, 5) = (7, 5).
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If these sides are vertical, they are parallel to the y-axis. Therefore, their y-coordinates change by 10 units, but their x-coordinates remain the same. The coordinates of the other two vertices are (-3, 12+10) = (-3, 22) and (-3, 5-10) = (-3, -5).
So, the points that could be a vertex of the rectangle are (7, 12), (7, 5), (-3, 22), and (-3, -5).
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