This exam was adminstered in August 2023.

### August 2023 Geometry, Part II

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

**25.** *
On the set of axes below, congruent quadrilaterals ROCK and R'O'C'K' are graphed.
*

*Describe a sequence of transformations that would map quadrilateral ROCK onto
quadrilateral R'O'C'K'.
*

**Answer: **

The figure has to move from Quadrant IV to Quadrant II. Inspection shows that the orientation of the quadrilateral has rotated. However, a rotation about the origin will not create the image of K at (-1,1), so a translation must follow.

Rotate ROCK 180 degrees around the origin. Then translate the image 2 to the left and 1 up.

There are many possible answers involding rotations around different centers, or reflections across the x-axis and y-axis or even other lines.

**26.** * In triangle CEM, CE = 3x + 10, ME = 5x - 14, and CM = 2x - 6.
Determine and state the value of x that would make △CEM an isosceles triangle with
the vertex angle at E.
*

**Answer: **

If the vertex angle is E then sides CE and ME must be congruent and their lengths are equal.

Therefore,

2x = 24

x = 12

**27.** *A flagpole casts a shadow on the ground 91 feet long, with a 53° angle of elevation from the end
of the shadow to the top of the flagpole.
*

*Determine and state, to the nearest tenth of a foot, the height of the flagpole.
*

**Answer: **

The height and the shadow mean that we need to use tangent. (We aren't concerned with the hypotenuse of the right triangle created.

The height is opposite the 53° angle and the shadow on the ground is adjacent to it.

x = 91 tan 53 = 120.76... = 120.8 feet

**28.** *A man is spray-painting the tops of 10 patio tables. Five tables have round tops, with diameters
of 4 feet, and five tables have rectangular tops, with dimensions of 4 feet by 6 feet. A can of spray
paint covers 25 square feet. How many cans of spray paint must be purchased to paint all of the
tabletops?
*

**Answer: **

Find the area of one circle (πr

^{2}) and multiply by 5. Find the area of one rectangle (A=LW) and multiply by 5.

A = 5π(2)^{2} + 5(4)(6) = 182.83

Divide this by 25 and round up to the next can: 182.83 / 25 = 7.3...

8 cans of paint are needed.

If you don't round up, you will not have enough paint to finish the job.

**29.** *Using a compass and straightedge, construct a midsegment of △AHL below.
[Leave all construction marks.]
*

**Answer: **

To construct a midsegment, you need two midpoints. To get two midpoints, you need to construct two perpendicular bisectors. You have a choice of which side you want to bisect. You may NOT use a ruler to measure the midpoints.

From point H, make an arc that is more than half the length of HL and swing it across HL and HA. (You could make two separate arcs, but there's no reason not to use the same one.

From point A, make an arc that intersects the first arc in two places. From point L, construct a third arc that intersect the arc drawm from point H.

Draw the two perpendicular bisectors. These will give you the midpoints. Use a straightedge to connect the midpoints. This is the midsegment, which is half the size of AL and parallel to AL.

**30.** *
Right triangle STR is shown below, with m∠T = 90°. Altitude TQ is drawn to SQR, and TQ = 8.
*

If the ratio SQ:QR is 1:4, determine and state the length of SR.

If the ratio SQ:QR is 1:4, determine and state the length of SR.

**Answer: **

Label SQ x and QR 4x. The product of (SQ)(QR) is equal to the square of (QT) by the Right Triangle Altitude Theorem.

(x)(4x) = 8^{2}

4x^{2} = 64

x^{2} = 16

x = 4

SR is x + 4x = 5x, and x = 4, so 5x = 5(4) = 20. SR = 20.

**31.** * Line AB is dilated by a scale factor of 2 centered at point A.
*

Evan thinks that the dilation of AB will result in a line parallel to AB, not passing through points A or B.

Nathan thinks that the dilation of AB will result in the same line, AB.

Who is correct?

*Explain why.
*

**Answer: **

Nathan is correct because every point on line AB will move twice as far from A in the same direction along the line AB. Also, since the image of point A will conincide with point A, parallel lines aren't possible.cimal because it's infinite.

**End of Part II**

How did you do?

Questions, comments and corrections welcome.

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