Geometry Problems of the Day (Geometry Regents, August 2010)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

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Geometry Regents, August 2010

Part IV: A correct answer will receive 6 credits. Partial credit is available.


38. Given: Quadrilateral ABCD has vertices A(−5,6), B(6,6), C(8,−3), and D(−3,−3).
Prove: Quadrilateral ABCD is a parallelogram but is neither a rhombus nor a rectangle.
[The use of the grid below is optional.]

Answer:

You don't need to graph the parallelogram, but if you do it will be "obvious" that it is neither a rectangle nor a rhombus. However, despite it being "obvious", you still have to PROVE it. (Seriously, writing "Look! It is isn't!" might get a laugh from the scorer, but no credit at all.) To show that ABCD is a a parallelogram, show that AB || CD and AD || BC. You can do this by finding the slopes of all four sides.

To show that ABCD is NOT a rectangle, show that two consecutive sides are not perpendicular -- you already have the slopes of the sides, so this is trivial.

To show that ABCD is NOT a rhombus, either show that two consecutive sides are not congruent, that is, they have different lengths, OR show that the slopes of the diagonals AC and BD are not perpendicular. Either is acceptable.

Slope of AB = (6-6)/(6-(-5)) = 0
Slope of BC = (-3-6)/(8-6) = -9/2
Slope of CD = (-3-(-3))/(-3-8) = 0
Slope of DA = (6-(-3))/(-5-(-3) = 9/(-2) = -9/2

The opposite sides have the same slopes, so the opposite sides are parallel. Since the opposite sides are parallel, the quadrilateral is a parallelogram.

Slope of AC = (-3 - 6)/(8 - (-5)) = -9/13
Slope of BD = (-3 - 6)/(-3 - 6) = -9/-9 = 1

Since AC is not perpendicular BD, then the diagonals of the parallelogram are not perpendicular, so the quadrilateral cannot be a rhombus.

Length of AB = √((-5-6)² + (6-6)²) = -11
Length of BC = √((6-8)² + (6-(-3))²) = √(4 + 81) = √(85)

The consecutive sides of the quadrilateral are not congruent, so the parallelogram cannot be a rhombus.

End of exam. How did you do?

More to come. Comments and questions welcome.

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