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 2011*

Part I: Each correct answer will receive 2 credits.

**6.** A straightedge and compass were used to create the construction
below. Arc EF was drawn from point B, and arcs with equal radii were
drawn from E and F.

Which statement is false?

1) m∠ABD = m∠DBC

2) 1/2 (m∠ABC) = m∠ABD

3) 2(m∠DBC) = m∠ABC

4) 2(m∠ABC) = m∠CBD

Which statement is false?

1) m∠ABD = m∠DBC

2) 1/2 (m∠ABC) = m∠ABD

3) 2(m∠DBC) = m∠ABC

4) 2(m∠ABC) = m∠CBD

**Answer: 4) 2(m∠ABC) = m∠CBD **

This is a construction of an angle bisector. So ADB = DBC, ABD is 1/2 of ABC, and ABC is twice DBC. But ABC is the large angle, so it would equal twice CBD, not the other way around.

Choice (4) is false.

**7.** What is the length of the line segment whose endpoints are (1,−4)
and (9,2)?

1) 5

2) 2 √(17)

3) 10

4) 2 √(26)

**Answer: 3) 10 **

Use the distance formula or Pythagorean Theorem. You may need to simplify a radical.

d = SQRT( (9-1)^{2} + (-4-2)^{2} )

= SQRT ( (8)^{2} + (-6)^{2} )

At this point, you should recognize that this will be the hypotenuse of a 6-8-10 triangle. If you didn't, continue with the calculations. You'll get 10.

**8.** What is the image of the point (2,−3) after the transformation r_{y-axis}?

1) (2,3)

2) (-2,-3)

3) (-2,3)

4) (-3,2)

**Answer: 2) (-2,-3) **

The original point is in Quadrant IV. If you flip it over the y-axis, it will end up in Quadrant III. The signs will both be negative but the magnitude of the numbers will remain the same.

In other words, the image will be at (-2,-3), which is Choice (2)?

**9.** In the diagram below, lines n and m are cut by transversals p and q.

Which value of x would make lines n and m parallel?

1) 110

2) 80

3) 70

4) 50

Which value of x would make lines n and m parallel?

1) 110

2) 80

3) 70

4) 50

**Answer: 3) 70 **

For the lines to be parallel, x would have to be congruent to the corresponding angle, which is the unmarked one between the 30 and the 80 degree angles.

Those three angles make a straight line so 30 + 80 + x = 180. Therefore, x = 70.

**10.** What is an equation of the circle with a radius of 5 and center
at (1,−4)?

1) (x + 1)^{2} + (y − 4)^{2} = 5

2) (x + 1)^{2} + (y − 4)^{2} = 5

3) (x + 1)^{2} + (y − 4)^{2} = 25

4) (x − 1)^{2} + (y + 4)^{2} = 25

**Answer: 4) (x − 1) ^{2} + (y + 4)^{2} = 25 **

The equation for a circle is (x − h)

^{2}+ (y - k)

^{2}= r

^{2}, where (h,k) is the center of the circle and r is the radius. There are minus signs in the formula, so the signs on the points are flipped.

Since the radius is 5, 5^{2} = 25. Eliminate Choices (1) and (2).

Since the signs get flipped, the correct choice is (4).

More to come. Comments and questions welcome.

More Regents problems.

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