Geometry Problems of the Day (Geometry Regents, August 2013)

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

More Regents problems.

Geometry Regents, August 2013

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


38.Quadrilateral ABCD with vertices A(-7,4), B(-3,6), C(3,0), and D(1,-8) is graphed on the set of axes below. Quadrilateral MNPQ is formed by joining M, N, P, and Q, the midpoints of AB, BC, CD, and AD, respectively.

Prove that quadrilateral MNPQ is a parallelogram.

Prove that quadrilateral MNPQ is not a rhombus

Answer:

Draw quadrilateral MNPQ. It will be help to see it visually even though you can do it algebraically.

The midpoints are (average of 2 x-coodinates, average of 2 y-coordinates).

They are M((-7-3)/2, (4+6)/2), N((-3+3)/2, (6+0)/2), P((3+1)/2, (0-8)/2), and Q((1-7)/2, (-8+4)/2),
Or M(-5, 5), N(0, 3), P(2, -4), and Q(-3, -2).
Again, you could've counted out the boxes on the graph if you hate formulas and averages or dealing with signed numbers in fractions.

To prove it's a parallelogram, you need to show that the opposite slopes are equal, which means that the opposite sides are parallel.

To prove that it NOT a rhombus, you can EITHER show that any two sides are not the same length, OR that the diagonals are not perpendicular to each other.

Find the Slopes of MN, NP, PQ, and QM:

MN: (3 - 5) / (0 - -5) = -2 / 5 = -2/5
NP: (-4 - 3) / (2 - 0) = -7 / 2 = -7/2
PQ: (-2 - -4) / (-3 - 2) = 2 / -5 = -2/5
QM: (5 - -2) / (-5 - -3) = 7 / -2 = -7/2

The opposite sides have the SAME slope, so the opposite sides are PARALLEL. Therefore, MNPQ is a Parallelogram.

Find the Slopes of diagonals MP and NQ:

MP: (-4 - 5) / (2 - -5) = -9 / 7
NQ: (-2 - 3) / (-3 - 0) = -5 / -3 = 5 / 3

The product of (-9/7)(5/3) =/= -1, so the diagonals are not perpendicular. Therefore, MNPQ is NOT a Rhombus.

Find the Lengths of sides:

MN: SQRT ((5)² + (2)²) = SQRT(29)
NP: SQRT((2² + (7)²) = SQRT(51)

The sides of the parallelogram are NOT congruent. Therefore, MNPQ is NOT a rhombus.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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