Geometry Problems of the Day (Geometry Regents, June 2025 Part IV)

This exam was adminstered in June 2025.

June 2025 Geometry, Part IV

Each correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Triangle PET has vertices with coordinates P(26,4), E(6,8), and T(24,22).
Prove △PET is a right triangle.
[The use of the set of axes on the next page is optional.]

State the coordinates of N, the image of P, after a 180° rotation centered at (1,3).

Prove PENT is a rectangle.
[The use of the set of axes below is optional.]

Answer:

Use the graph. First, it counts as work. Second, having a visual representation makes solving the problem easier. Third, if the triangle is graphed, you can count boxes rather than use formuals and risk encountering a sign or other computational error.

The slope of PE is 4/12 = 1/3.

The slope of PT is -6/2 = -3.

The product of (1/3)(-3) = -1, so PE is perpendicular to PT. Therefore PET is a right triangle.

To find the coordinates of N, count the change in the x-value and the y-value from P to the coordinates (1,3). Then count those same numbers again. (Seven right and one down.) The coordinates of N are (8,2).

The slope of EN is -6/2 = -3.

The slope of TN is 4/12 = 1/3.

PE || NT and PT || EN, so PENT is a parallelogram. Angle P is a right angle, so PENT is a rectangle.

End of Part IV

End of Exam

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