August 2018 Common Core Geometry Regents, Part II (open-ended)

The following are some of the multiple questions from the recent August 2018 New York State Common Core Geometry Regents exam.
The answers to Part I can be found here
The answers to Parts III and IV can be found here

August 2018 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

25. In the circle below, is a chord. Using a compass and straightedge, construct a diameter of the circle. [Leave all construction marks.]

Answer:
The diameter will be a perpendicular bisector of the chord, so you need to construct a perpendicular bisector of AB.
Open the compass more than half the width of AB. From point A, make small arcs on either side of the line. Do the same from point B without changing the width of the compass. Make points where the arcs intersect. Use the straightedge to draw the diameter. Make sure you draw the line long enough to intersect with the circle twice! You were asked for a diameter.

26. In parallelogram ABCD shown below, the bisectors of ∠ABC and ∠DCB meet at E, a point on AD.

If m∠A = 68°, determine and state m∠BEC.

Answer:
If m∠A = 68°, then m∠BCD = 68° and m∠ABC = (180 - 68)° = 112°.
Since BE and CE are bisectors, then m∠EBC = (112/2) = 56° and m∠BCE = (68/2) = 34°
We know 2 of the angles of triangle BEC, and the three of them add up to 180.
56 + 34 + m∠BEC = 180
m∠BEC = 90

A shorter, more generic version of this explanation:
Angles ABC and BCD are supplementary. They have a sum of 180 degrees.
Because BE and CE are bisectors, the sum of EBC and BCE must be half as much, which is 180/2 = 90 degrees.
Triangle BCE has a total of 180 degrees, so Angle BEC must be 180 - 90 = 90 degrees.
This will always be true regardless of the measure of angle A.

27. In circle A below, chord BC and diameter DAE intersect at F.

If mCD = 46° and mDB = 102°, what is m∠CFE?

Answer:
Note that DAE is a diameter. So the sum of arcs DB and BE is 180 degrees. That means that BE = 180 - 102 = 78 degrees.
You have a choice now, both of which use essentially the same steps, in a different order.
You can find the measure of angle BFE (and CFD), which is then supplementary to CFE, OR you can find the measure of arc CE and then find the measure of CFE.

Method 1: The measure of Angle BFE is the average of arcs CD and BE.
(46 + 78) / 2 = 124 / 2 = 62.
180 - 62 = 118.
Angle CFE is 118 degrees
Method 2: 46 + 102 + 78 + CE = 360
226 + CE = 360
CE = 134
(102 + 134) / 2 = 236 / 2 = 118
Angle CFE is 118 degrees.

28. Trapezoids ABCD and A"B"C"D" are graphed on the set of axes below.

Describe a sequence of transformations that maps trapezoid ABCD onto trapezoid A"B"C"D".

Answer:
There are multiple answers. Notice that points have rotated, so it is not a simple reflection.
Also notice that is not a simple rotation about the origin because AB is closer to the x-axis than A"B".

Some possible answers:
1: ABCD is rotated 180 degrees about the origin, followed by a translation T_{0, -2}.

2: ABCD is reflected across y = -1, followed by a reflection over the y-axis. (This could be in either order.)

3: ABCD is reflected across the x-axis, followed by a translation 2 units down, followed by a reflection over the y-axis.

4: ABCD is rotated 180 degrees about the point (0, -1).

29. In the model below, a support wire for a telephone pole is attached to the pole and anchored to a stake in the ground 15 feet from the base of the telephone pole. Jamal places a 6-foot wooden pole under the support wire parallel to the telephone pole, such that one end of the pole is on the ground and the top of the pole is touching the support wire. He measures the distance between the bottom of the pole and the stake in the ground.

Jamal says he can approximate how high the support wire attaches to the telephone pole by using similar triangles. Explain why the triangles are similar.

Answer:
The triangles are similar because the share the angle between the wire and the ground, and they both have vertical sides which are perpendicular to the ground, creating right angles. But the AA Postulate, the triangles are similar.

30. Aliyah says that when the line 4x + 3y = 24 is dilated by a scale factor of 2 centered at the point (3,4), the equation of the dilated line is y = -(4/3)x + 16. Is Aliyah correct? Explain why. [The use of the set of axes below is optional.]

Answer:
When a line is dilated, one of two things will happen: either you will get a parallel line, or the line doesn't change because the center point is on the line itself.
First thing, check if (3, 4) is a solution to 4x + 3y = 24
4(3) + 3(4) = 24
12 + 12 = 24
24 = 24
The dilated line is the same as the original line.

For Aliyah to be correct, y = -(4/3)x + 16 must be the same line as 4x + 3y = 24.
4x + 3y = 24
3y = -4x + 24
y = -(4/3)x + 8.
Aliyah is not correct.

31. Ian needs to replace two concrete sections in his sidewalk, as modeled below. Each section is 36 inches by 36 inches and 4 inches deep. He can mix his own concrete for $3.25 per cubic foot.

How much money will it cost Ian to replace the two concrete sections?

Answer:
Convert the inches into feet by dividing by 12.
The Volume of concrete he needs is 2 * 3 * 3 * (1/3) = 6 cubic feet.
Multiply the number of cubic feet by the cost per cubic foot:
6 * 3.25 = $19.50.

End of Part II

How did you do?

Questions, comments and corrections welcome.