Geometry Problems of the Day (Geometry Regents, January 2011)

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, January 2011

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


35. Quadrilateral MATH has coordinates M(1,1), A(−2,5), T (3,5), and H(6,1). Prove that quadrilateral MATH is a rhombus and prove that it is not a square.
[The use of the grid on the next page is optional.]

Answer:

Graphing the points could help by giving you a visual reference point but it isn't necessary.

A rhombus has four congruent sides. You have show this to be true by using the distance formula four times.

Or you can show that the rhombus is a parallelogram by finding the slopes of the four sides and then finding the lengths of two consecutive sides with the distance formula. Why? Because finding slopes is easier as less likely to cause an arithmetic error.

HOWEVER, you can ALSO show that a quadrilateral is a rhombus if its diagonals are perpendicular to each other. This is actually the simplest method, but it is not the most obivous.

The slope of MT is (5-1)/(3-1) = 4/2 = 2. The slope of AH is (1-5)/(6-(-2)) = -4/8 = -1/2. MT is perpendicular to AH, so the quadrilateral is a rhombus.

To show that it is not a square, show that the sides of the rhombus are not perpendicular.

The slope of MA is (5-1)/(-2-1) = 4/-3. The slope of AT is (5-5)/(3-(-2)) = 0/5 = 0. The sides are not perpendiclar, so the rhombus is not a square.

Note that the second part would be obvious from a graph because two lines would be horizontal, but the other two lines would not be vertical.

If you use the distance formula for the first part, you would have found that:

MA = SQRT( (1-(-2)² + (1-5)² ) = SQRT(9 + 16) = 5

AT = SQRT( ((-2)-3² + (5-5)² ) = SQRT(25 + 0) = 5

TH = SQRT( (3-6)² + (5-1)² ) = SQRT(9 + 16) = 5

AT = SQRT( (6-1)² + (1-1)² ) = SQRT(25 + 0) = 5

All sides are congruent.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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