**New York State Geometry Regents exam**. The open-ended problems will (most likely) appear tomorrow. Since I typed this up as quickly as I could, there are no images, graphs or diagrams included. I'll edit them as I am able to.

A common theme: there were several questions involving the **equation of a circle**, as there always are, as well as several questions which not only involved the **Pythagorean Theorem**, but very specifically a **3-4-5** triangle or a multiple of it.

I hope everyone did well. Part I was, in my opinion, very "get-able" -- not very difficult for a Regents exam.

As always, apologies in advance for typos. I did rush this a bit. Here is Part I.

*1.**
What is the solution of the system of equations graphed below? (Image
omitted)*

*y = 2x
+ 1*

*y = x ^{2}
+ 2x – 3*

The
solutions of a system are the points where the two lines intersect. In this
problem, that would be at (-2, -3) and (2, 5). Choice (4).

If you
look at the other choices, (0, -3) is the y-intercept of the parabola, (-1, -4)
is the vertex and (-3,0) and (1, 0) are the roots of
the parabola. None of those have anything to do with the linear equation.

*2.** What are the coordinates of the midpoint of
the line segment with endpoints (2, -5) and (8, 3)?*

Take
the average of the x-coordinates and of the y-coordinates. You get (5, -1) which is exactly in the
middle. Alternatively, you could have used the scratch graph paper and plotted
it instead of calculating.

*3.** As** shown in the diagram below, when hexagon
ABCDEF is reflected over line m, the image is hexagon A’B’C’D’E’F’. Under this
transformation, which property is not preserved? (Image omitted)*

Orientation
is not preserved. The object has the same shape and size, but the direction its facing switches.

*4.** In** the diagram of triangle ABC below, BD is
drawn to side AC. (Image omitted)*

*If m<A = 35, m<ABD = 25, and m<C =
60, which type of triangle is BCD?*

BCD is
equilateral. Angles A and ABD add up to 60 degrees. This makes <ADB = 120
degrees. Angle BDC is supplementary to ADB, so it is 60 degrees. Angle C is 60
degrees as well, so <CBD must be 60 degrees. Triangle BCD has three
60-degree angles, so it is an *equilateral*
triangle.

*5.** In the diagram
below of rhombus ABCD, the diagonals AC and BD intersect at E. (Image omitted)*

*If AC = 18 and BD = 24, what is the length
of one side of rhombus ABCD?*

The
diagonals of a rhombus are perpendicular and bisect each other. They form four
congruent right triangles. Each side of the rhombus is the hypotenuse of one of
those right triangles. AC = 18, so AE = 9. BD = 24, so BE = 12. If the two legs
of a right triangle are 9 and 12, then the hypotenuse is 15. *Note: You can use the Pythagorean Theorem, but you should know 9-12-15 because it’s a
multiple of 3-4-5, which is the most common right triangle used in these
problems!* The answer is 15.

*6.** What are the truth values of the statement
“Opposites angles of a trapezoid are always congruent.” and its negation?*

The statement
is false, which makes the negation (“Opposite angles of a trapezoid are not
always congruent.” ) true.

*7.** What is the length of a line segment whose
endpoints have coordinates (5, 3) and (1, 6)?*

Use the
distance formula, or sketch the line segment, make a right triangle and use
Pythagorean Theorem. Either way, you will get the square root of (3^{2}
+ 4^{2}), which is the square of (9 + 16), which is
5. Again, 3-4-5 is a common right triangle. Recognize it when you see it!

*8.** In** the diagram below of isosceles triangle
ABC, the measures of vertex angle B is 80 degrees. If AC extends to point D,
what is m<BCD? (Image omitted)*

The
vertex angle is 80 degrees, so the two base angles have a total of 100 degrees.
The base angles are congruent, so each is 50 degrees. Angle BCD is supplementary to a 50 degree
angle, so it is 130 degrees.

*9.** A student used a compass and a
straightedge to construct CE in triangle ABC as shown below. (Image omitted)*

*Which statement must always be true for
this construction? *

The
construction shows the bisecting of angle C. So <ACE is congruent to <
BCE.

*10.** In triangle, ABC,
AB = 4, BC = 7, and AC = 10. Which statement is true?*

The
smallest angle is across from the smallest side and the biggest angle is across
from the biggest side. (Triangle Inequality Theorem).
The opposite angle will be the vertex that is NOT part of the line segment;
i.e., <C is across from AB and is the smallest angle, and <B is the
largest. The correct choice is (2)
m<B > m <A > m< C. (Sorry
about the formatting. It can’t be helped.)

*11.** A circle whose center has coordinates (-3,
4) passes through the origin. What is the equation of this circle?*

Using
Pythagorean Theorem, we can see that the radius of the circle is 5. *Again, another 3-4-5 triangle!*

Flip
the signs and square the radius: (x + 3)^{2} + (y – 4)^{2} = 25

*12.** Point W is located in plane R. How many
distinct lines passing through point W are perpendicular to plane R?*

One. Think of any vertical pole coming out of the
ground. There’s only one possible vertical pole at that point in the ground.

*13.** In the diagram below (image omitted), line
l is parallel to line m, and line w is a transversal.*

*If m<2 = 3x + 17 and m <3 = 5x – 21,
what is m<1?*

Angles
2 and 3 are supplementary. Angles 1 and 2 are congruent. Solve for x and then
find the measure of angle 2 to get the answer.

3x + 17 + 5x – 21 = 180

8x – 4 = 180

8x = 184

x = 23

m<2 = 3(23) + 17 = 69 + 17 = 86. Choice
(4).

*Note: be careful. I subtracted 4 instead of
adding first time through. My incorrect answer wasn’t one of the choices, so I
knew to go back and check my work!*

*14.** The diagram below is of circle O. (Image
omitted – it has a circle with center O at (5, -3) and a radius of 4.)*

*Which equation represents circle )?*

*Again.* Flip
the signs and square the radius. (x – 5)^{2} + (y +
3)^{2} = 16.

*15. **In** isosceles trapezoid QRST show below, QR
and TS are bases. (Image omitted)*

*If m<Q = 5x + 3 and m<R = 7x – 15,
what is m<Q?*

Angles
Q and R are congruent, so their measures are equal.

5x + 3 = 7x – 15

3 = 2x – 15

18 = 2x

9 = x

m<Q = 5(9) + 3 = 45 + 3 = 48. Choice
(2).

*16.** Triangle ABC is graphed on the set of axes
below. (Image omitted)*

*What are the coordinates of the point of
intersection of the medians of triangle ABC?*

This
one is easier than it sounds. The median from vertex A to the midpoint of BC is
a horizontal line with a length of 6. The medians will meet at the centroid,
which is 2/3 of the way from A to the midpoint. That means that it will be 4
units to the right of point A. The answer is (-1, 2).

If you
drew the three medians (two would be enough, though), you could probably
eyeball the correct answer from the choices given. Only one really makes sense.

*17.** Given the statement, “If a number has
exactly two factors, it is a prime number,” what is the contrapositive of this
statement?*

Negative
both parts and flip it around. “If a number is not a prime number, it does not
have exactly two factors.” Choice (2).

*18.** Which graph represents a circle whose
equation is (x – 2) ^{2} + (y + 4)^{2} = 4. (Images omitted)*

Third
time! Flip the signs. The center is (2, -4). The radius is 2, which is the
square root of 4. That makes Choice (3) the correct answer.

*19.** If two sides of a triangle have lengths of
4 and 10, the third side could be …*

The
third side must be greater than 6, which is 10-4, and less than 14, which is
10+4. The only choice that fits is Choice (1) 8.

*20.** The lines represented by the equations 4x
+ 6y = 6 and y = 2/3x – 1 are …*

Put the
first line into slope-intercept form:

4x + 6y
= 6

6y =
-4x + 6

y =
-2/3x + 1

The
lines have the different slope, so they are neither parallel nor the same line.
Additionally, the slopes are not negative reciprocals, so they are not
perpendicular. However, they will intersect.

*21.** In the diagram
below of triangle ABC, DE || AB. (Image omitted)*

*If CD = 4, CA = 10, CE = x + 2 and EB = 4x
– 7, what is the length of CE?*

Set up
a proportion. CD/DA = CE/EB. Notice that
I said DA, not CA. DA = 10-4 = 6.

So 4/6 = x+2/4x -7. Cross multiply add you get

4(4x – 7) = 6(x + 2)

16x – 28 = 6x + 12

10x = 40

X = 4

CE = 4 + 2 = 6. Choice (3).

*22.** Parallelogram ABCD with diagonals AC and
BD intersecting at E is shown below (image omitted).*

*Which statement must be true?*

Diagonals
of a parallelogram bisect each other, but that does not appear in the choices.

Each
diagonal is a transversal across two parallel lines, so the alternate interior
angles are congruent. This is needed to prove choice (2).

*23.** In the diagram
below of circle O, m<ABC = 24. (Image omitted)*

*What is m<AOC?*

The
inscribed angle is half the size of the central angle. If m<ABC is 24, then
m<AOC = 48.

*24.** Triangle A’B’C’ is the image of triangle
ABC after a dilation of 2. Which statement is true?*

The
original triangle will have sides that are half as long as the image. The
angles will be the same. So the angle is choice (3), m<B = m<B’.

*25.** In the diagram of
the circle below, AD || BC, arc AB = (5x + 30) degrees, and arc CD = (9x – 10)
degrees. (Image omitted)*

*What is mAB?*

If the
chords are parallel, then the arcs they create are congruent.

9x – 10 = 5x + 30

4x = 40

X = 10

AB =
5(10) + 30 = 50 + 30 = 80. Choice (4).

*26.** The bases of a
prism are right trapezoids, as shown in the diagram below. (Image omitted)*

*Which two edges do not lie in the same
plane?*

Choice
(1), BC and WZ. You cannot find a piece of paper which lines up
with both edges. They are skew.

*27.** In the diagram
below, A’B’ is the image of AB under which single transformation? (Image
omitted)*

It’s
flipped over the x-axis and slid back a few spaces. It’s a glide reflection.

*28.** For which diagram
is the statement triangle ABC ~ triangle ADE not always true?*

Choice
(4), which shows a trapezoid, with A as the midpoint of the diagonals.
Triangles ABC and ADE share no angles and have none which must be congruent.
Therefore, they are not always similar.