January 2024 Geometry Regents Part IV

This exam was adminstered in January 2024.

January 2024 Geometry, Part IV

A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.

35. Quadrilateral MATH has vertices with coordinates M(-1,7), A(3,5), T(2,-7), and H(-6,-3).
Prove that quadrilateral MATH is a trapezoid.
[The use of the set of axes on the next page is optional.]

State the coordinates of point Y such that point A is the midpoint of MY.

Prove that quadrilateral MYTH is a rectangle. [The use of the set of axes below is optional.]

Answer:

You don't neeed to use the grid but it may be helpful for visualizing. If you do use it, you still need to answer the questions fully and completely. You can't rely on whatever is in the grid to be sufficient.

To show that MATH is a trapezoid, you have to that there is one pair of parallel sides. You can do this by find the slopes of the four sides. This is easy if you graph it because you can count boxes without worrying about subtracting signed numbers (if that's a problem for you). Only two of the sides will be the same and the other two will be different.

Slope MA (7 - 5) / (-1 - 3) = 2/-4 = -1/2

Slope AT = (-7 - 5) / (2 - 3) = -12/-1 = 12

Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2

Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Since there is one pair of parallel sides, the quadrilateral is a trapezoid.

To find point Y, find the change in x-value and y-value fro M to A and add those numbers again to get Y.

M(-1,7) -> A(3,5) is a translation of +4,-2. Y(3+4,5-2) = Y(7,3)
This was worth a point even if you didn't show work.

To show that MYTH is a rectangle, you could show that the opposite sides are parallel (same slope) and that two consecutive sides are parallel (slopes are inverse reciprocals).

Slope MY (7 - 3) / (-1 - 7) = 4/-8 = -1/2

Slope YT = (-7 - 3) / (2 - 7) = -10/-5 = 2

You previously found:
Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2
Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Opposite sides are parallel, so it is a parallelogram.

(-1/2)(1) = -1. MY is perpendicular to YT, so angle Y is a right angle. Therefore, MYTH is a rectangle.

End of Exam

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