The following are some of the multiple questions from the recent June 2018 New York State Common Core Geometry Regents exam.

The answers to Part I can be found here

### June 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.** *
Triangle A'B'C' is the image of triangle ABC after a translation of 2 units to the right and
3 units up. Is triangle ABC congruent to triangle A'B'C'? Explain why
*

**Answer: **

Yes, because a translation is a rigid motion, which preserves shape and size.

The corresponing angles are congruent, and the corresponding sides are congruent, so the triangles are congruent.

**26.** *
Triangle ABC and point D(1,2) are graphed on the set of axes below.
*

*See image below.*

Graph and label triangle A'B'C', the image of triangle ABC, after a dilation of scale factor 2 centered at point D.

Graph and label triangle A'B'C', the image of triangle ABC, after a dilation of scale factor 2 centered at point D.

**Answer: **

See image below. No explanation is needed. Just the graph, with labels.

Each image is twice as far away from D as the original point was.

To get from D to A, go down 1 unit and 3 left. To get from A to A', do this again. A' is (-5, 0)

To get from D to B, go up 3 units and 1 left. To get from B to B', do this again. B' is (-1, 8)

To get from D to C, go down 3 units and 3 right. To get from C to C', do this again. C' is (7, -4)

**27.** *
Quadrilaterals BIKE and GOLF are graphed on the set of axes below.
*

Describe a sequence of transformations that maps quadrilateral BIKE onto quadrilateral GOLF.

Describe a sequence of transformations that maps quadrilateral BIKE onto quadrilateral GOLF.

**Answer: **

Notice that letters of BIKE go counterclockwise and GOLF go clockwise. A simple translation will not be good enough. A translation of T_{5,8} will map BIKE over GOLF; however, B doesn't map to G, and K doesn't map to L.

You need to include a reflection in your transformations. The directions ask for a sequence.

There are many possibilities, but the simplest (to me) would be a reflection over the y-axis followed by a translation of 5 units up.

**28.** *
In the diagram below, secants RST and RQP, drawn from point R, intersect circle O at S, T, Q, and P.
*

If RS = 6, ST = 4, and RP = 15, what is the length of RQ?

If RS = 6, ST = 4, and RP = 15, what is the length of RQ?

**Answer: **

(RS)(RT) = (RQ)(RP)

(6)(6 + 4) = (RQ)(15)

60 = 15(RQ)

RQ = 4.

**29.** *
Using a compass and straightedge, construct the median to side AC in ABC below.
[Leave all construction marks.]
*

**Answer: **

See image.

To construct a median, you need to find the midpoint of AC. You can find the midpoint by constructing a perpendicular bisector of AC. Label the midpoint D. Then use the straightedge to draw BD.

**30.** *
Skye says that the two triangles below are congruent. Margaret says that the two triangles are
similar.
Are Skye and Margaret both correct? Explain why.
*

**Answer: **

Skye and Margaret are both correct. Using the Pythagorean Theorem, 5^{2} + 12^{2} = 169, which is 13^{2}, so the length of the hypotenuse of the first triangle is 13. By the Hypotenuse-Leg Theorem, the two triangles are congruent.

If the triangles are congruent, then their corresponding angles are congruent, and that makes them similar triangles as well.

**31.** *
Randy’s basketball is in the shape of a sphere with a maximum circumference of 29.5 inches.
Determine and state the volume of the basketball, to the nearest cubic inch.
*

**Answer: **

The circumference is 2*pi*r = 29.5

So the radius, r = 29.5 / (2 * pi) = 4.695...

Volume of a sphere = (4/3) * pi * r^{3} = (4/3) * pi * (4.695)^{3} = 433.506...

To the nearest cubic inch, the Volume is 434 cubic inches.

**End of Part II**

How did you do?

Questions, comments and corrections welcome.

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