Yahto and Nora want to find out if the quadrilateral formed by connecting points J, K, L, and M is a rectangle.

Yahto’s plan:
• Use the distance formula to find the lengths of the sides.
• Then see if the side lengths of opposite sides are the same.

Nora’s plan:
• Find the slopes of the four sides.
• See if the slopes of adjacent sides are negative reciprocals of each other.

Which student’s plan will work?

A. Both plans are correct.

B. Neither plan is correct.

C. Only Yahto’s plan is correct.

D. Only Nora’s plan is correct.

Yahto has the right idea in that opposite sides of a rectangle are the same length. However, that is also true of a parallelogram that is not a rectangle. The condition Yahto is looking for is necessary, but not sufficient.

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Nora's plan will discover if adjacent sides are perpendicular to each other, provided that the slopes are defined in each case. Her plan will not work in the event there are vertical sides with undefined slope. Any quadrilateral in which all adjacent pairs of sides form right angles will be a rectangle.

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Strictly speaking, neither plan is completely correct. Yahto can discover rectangles that Nora cannot, and Nora can determine quadrilaterals are not rectangles when Yahto would improperly classify them.

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Comment on showing a quadrilateral is a rectangle

My favorite plan is to show the diagonals are the same length and have the same midpoint: J+L = K+M; ║J-L║ = ║K-M║.