Saturday, October 09, 2021

Geometry Problems of the Day (Geometry Regents, August 2012)

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 2012

Part I: Each correct answer will receive 2 credits.

1. In the diagram below of circle O, chord AB is parallel to chord GH. Chord CD intersects AB at E and GH at F.

Which statement must always be true?

1) Arc AC ≅ arc CB
2) Arc DH ≅ arc BH
3) Arc AB ≅ arc GH
4) Arc AG ≅ arc BH

Answer: 4) Arc AG ≅ arc BH

If two chords are parallel, then the arcs that they intercept are congruent.

Choice (1) would only be true if CD was a diameter.

In Choice (2), there is nothing special about the pair of arcs or the chords that create them.

Choice (3) is a common misunderstanding even though it is plainly obvious that it isn't true.

2. The vertices of parallelogram ABCD are A(2,0), B(0,-3), C(3,-3), and D(5,0). If ABCD is reflected over the x-axis, how many vertices remain invariant?

1) 1
2) 2
3) 3
4) 0

Answer: 2) 2

Invariant means that the point doesn't move when the parallelogram is reflected. The only way for that to happen is if the vertices were on the x-axis (or whatever line the pre-image is to be reflected over).

Points A and D are on the x-axis. Point B is on the y-axis. Point C is on the line y = x.

The answer is two points, A and D.

104 + 168 + x + x = 360
272 + 2x = 360
2x = 88
x = 44

3. Point M is the midpoint of AB. If the coordinates of A are (-3,6) and the coordinates of M are (-5,2), what are the coordinates of B?

1) (1,2)
2) (7,10)
3) (-4,4)
4) (-7,-2)

Answer: 4) (-7,-2)

Look at the change in the coordinates from point A to point M. Apply those changes to point M to get point B.

-5 - -3 = -2 and 2 - 6 = -4

Point B is (-5 - 2, 2 - 4), or (-7, -2)

4. When a dilation is performed on a hexagon, which property of the hexagon will not be preserved in its image?

1) parallelism
2) orientation
3) length of sides
4) measure of angles

Answer: 3) length of sides

By definition, a dilation will increase or decrease the size of the pre-image. So the length of the sides will NOT remain constant.

A dilation will retain the orientation and the shape of the original. Since the shape is similar, the size angles will be the same and lines that were parallel will remain parallel.

5. As shown in the diagram below of triangle ABC, a compass is used to find points D and E, equidistant from point A. Next, the compass is used to find point F, equidistant from points D and E. Finally, a straight-edge is used to draw ray AF. Then, point G, the intersection of ray AF and side BC of triangle ABC, is labeled.

Which statement must be true?

1) AF bisects side BC
2) AF bisects ∠BAC
3) AF ⊥ BC
4) triangle ABG ~ triangle ACG

Answer: 2) AF bisects ∠BAC

These are the instructions for constructing an angle bisector, so the answer is Choice (2).

In Choice (1), the image shows that neither points B nor C were used for the construction, so AF wasn't made to bisect it. Similarly for Choice (3), there is no way to constuct AF perpendicular to BC if the were no marks starting from B or C.

In Choice (4), only angles DAG and EAG are congruent. There is nothing to say that angles B and C are congruent, or that AF is perpendicular to BC to create two right angles.

More to come. Comments and questions welcome.

More Regents problems.

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