Geometry Problems of the Day (Geometry Regents, January 2013)

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, June 2013

Part I: Each correct answer will receive 2 credits.


1. If triangle MNP ≅ triangle VWX and PM is the shortest side of triangle MNP, what is the shortest side of VWX?

1) XV
2) WX
3) VW
4) NP

Answer: 1) XV

If two triangles are congruent, then the corresponding sides are congruent. When listing two congruent triangles, the order of the letters matters. In this example, angle M corresponds to angle V, etc.

2. In circle O shown in the diagram below, chords AB and CD are parallel.

If m AB = 104 and m CD = 168, what is m BD ?

1) 38
2) 44
3) 88
4) 96

Answer: 2) 44

Arcs AC and BD are equal in size because the chords are parallel. The FOUR chords together add up to 360 degrees. Don't forget the fourth chord.

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

3. As shown in the diagram below, CD is a median of ABC.

Which statement is always true?

1) AD ≅ DB
2) AC ≅ AD
3) ∠ACD ≅ ∠CDB
4) ∠BCD ≅ ∠ACD

Answer: 1) AD ≅ DB

The definition of median is that D will be the midpoint of AB, so AD ≅ DB

Choice (2) could be true but usually will not be.

Choice (3) can never be true because ∠CDB is an exterior angle and is the sum of ∠ACD and ∠DAC.

Choice (4) would be true if CD were also an angle bisector, but it is only a median.

4. In the diagram below, under which transformation is ABC the image of ABC?

1) D₂
2) r_x-axis
3) r_y-axis
4) (x, y) --> (x-2, y)

Answer: 3) r_y-axis

The preimage was flipped over the y-axis to get the image. So it is a reflection.

Choice (1) is a Dilation of scale 2 would have doubled the size. This didn't happen.

Choice (2) is a reflection over the x-axis, which would have flipped the image upside down.

Choice (4) would move the preimage two units to the left.

5. Line segment AB is a diameter of circle O whose center has coordinates (6,8). What are the coordinates of point B if the coordinates of point A are (4,2)?

1) (1,3)
2) (5,5)
3) (8,14)
4) (10,10)

Answer: 3) (8,14)

If the center is point O, (6, 8), then the distance from the point O to B is the same as the distance from point O to A. More to the point, point O is the midpoint of AB.

To get from A to O, you go 6 - 4 = 2 units to the right and 8 - 2 = 6 units up.

Moving (+2, +6) from O means that B is (6+2, 8+6) = (8, 14).

More to come. Comments and questions welcome.

More Regents problems.

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