January 2023 Geometry Regents, Part IV

This exam was adminstered in January 2023. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

January 2023 Geometry, Part IV

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

35. Given: Triangle DUC with coordinates D(-3,-1), U(-1,8), and C(8,6)

Prove: ∆DUC is a right triangle
[The use of the set of axes on the next page is optional.]

Point U is reflected over DC to locate its image point, U', forming quadrilateral DUCU9. Prove quadrilateral DUCU' is a square.

Answer:
To show that DUC is a right triangle, find the slopes of all three sides. Two of them will be negative reciprocals, which means that the lines are perpendicular, and create a right angle.

To show that the quadrilateral is a square, find the slopes of the new lines to show that the figure is a parallelogram. The right angle makes it a rectangle. Find the length of of two consecutive sides to show that it is a square.

Slope of DC = (6 - -1)/(8 - -3) = 7/11

Slope of DU = (8 - -1)/(-1 - -3) = 9/2

Slope of UC = (6 - 8)/(8 - -1) = -2/9. DU is perpendicular to UC.

Therefore DUC is a right triangle.

Graph the three points to find U'.

What did I do in this image?

I drew a line from U that was perpendicular to CD. Since the slope fo CD is 7/11, then the slope of UU' must be -11/7. I plotted a point that was 11 units down and 7 units to the right and drew the line.

Now look at the distance from U to CD. You can see that it is half the distance from U to the new point. So this new point is U'. If you don't believe me, you can use the distance formula.

U' is located at (6,-3).

Now draw the quadrilateral DUCU'. Find the slopes of DU' and CU'.

Slope of DU' = (-3 - -1) / (6 - -3) = -2/9. DU' is parallel to UC.

Slope of CU' = (-3 - 6) / (6 - 8) = -9/-2 = 9/2. CU' is parallel to DU.

DUCU' is a parallelogram. Since UC is perpendicular to DU, DUCU' is a rectangle.

Show that DU = UC and you are done.

Length of DU = √(2² + 9²) = √(85).

Length of UC = √(9² + 2²) = √(85).

DU = UC so consecutive sides of a rectangle are congruent so the rectangle is a square.

Alternate method: If the rectangle is a square then the diagonals are congruent and perpendicular. We already showed that they are perpendicular, with slopes 7/11 and -11/7. You only have to find the lengths of the diagonals.

Length of DC = √(11² + 7²) =√(170).

Length of UU' = √(7² + 11²) =√(170).

The diagonals are perpendicular and congruent so the rectangle is a square.

End of Part Eaxm

How did you do?

Questions, comments and corrections welcome.

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