June 2018 Common Core Geometry Regents, Parts III and IV

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
The answers to Part II can be found here

June 2018 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

32. Triangle ABC has vertices with coordinates A(-1,-1), B(4,0), and C(0,4). Prove that ABC is an isosceles triangle but not an equilateral triangle. [The use of the set of axes below is optional.]

Answer:
If it is isosceles, then at least two legs have the same length. If it is not equilateral, then the third leg will have a different length.
Looking at the coordinates of the points, it should be obvious that AB and AC are congruent because you'll be using the same numbers in the calculations.
AB = SQRT ( (-1 - 4)² + (-1 - 0)² ) = SQRT(26)
AC = SQRT ( (-1 - 0)² + (-1 - 4)² ) = SQRT(26)
BC = SQRT ( (4 - 0)² + (0 - 4)² ) = SQRT(32)
There is no need to simplify because you're only looking for equality or inequality.
AB is the same length as AC but not the same as BC. Only two sides are congruent, so ABC is isosceles but not equilateral.

33. The map of a campground is shown below. Campsite C, first aid station F, and supply station S lie along a straight path. The path from the supply station to the tower, T, is perpendicular to the path from the supply station to the campsite. The length of path FS is 400 feet. The angle formed by path TF and path FS is 72°. The angle formed by path TC and path CS is 55°.

Determine and state, to the nearest foot, the distance from the campsite to the tower.

Answer:
We can find the length of TS by using the tangent function with triangle TSF.
Once we know TS, we can use the sine function with triangle TSC to find the length of CT, the distance from the campsite to the tower.
Make sure your calculator is in Degree mode.

Tan 72 = x / 400
x = 400 * tan 72 = 1231.07...

Sin 55 = 1231.07 / y
y = 1231.07 / sin 55 = 1502.85897... = 1503 feet.
The distance from the campsite to the tower is 1503 feet.

Note that you could have skipped the intermediary skip and used (x / 400) in the last equation. Finding the length of TS was not required for the problem.

34. Shae has recently begun kickboxing and purchased training equipment as modeled in the diagram below. The total weight of the bag, pole, and unfilled base is 270 pounds. The cylindrical base is 18 inches tall with a diameter of 20 inches. The dry sand used to fill the base weighs 95.46 lbs per cubic foot.

To the nearest pound, determine and state the total weight of the training equipment if the base is filled to 85% of its capacity.

Answer:
Find the Volume of the Base in cubic feet (not cubic inches). Multiply that by .85 to find 85% of the Volume. Multiply that by 95.46 to find the weight of the sand in the base. Then add 270 pounds for the bag, pole and unfilled base.
Remember that the radius is half the diameter: 20 / 2 = 10 inches, which is 10/12 of a foot. The height is 18 inches, which is 18/12 feet.
V = pi * r² * h = (3.141592...)(10/12)² * 18/12 = 3.27249
.85V = .85 * 3.27249 = 2.7816 cubic feet.
The weight of the sand = 2.7816 * 95.46 = 265.53 pounds.
Total weight = 265.53 + 270 = 535.53 or 536 pounds.

End of Part III

Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. Parallelogram ABCD, BF ⊥ AFD, and DE ⊥ BEC.

Prove: BEDF is a rectangle

Answer:
BEDF is a rectangle if it is a parallelogram as has a right angle. You need not prove the length of the opposite sides are congruent.
1. ABCD is a parallelogram, BF ⊥ AFD, DE ⊥ BEC: Given
2. BC || AD: Opposite sides of a parallelogram are parallel.
3. BE || DE: Parts of parallel lines are parallel.
4. BF || DE: Two lines that are perpendicular to the same line are parallel.
5. BEDF is a parallelogram: A quadrilateral with two pairs of parallel sides is a parallelogram.
6. Angle DEB is a right angle: Perpendicular lines form right angles.
Note: Line 6 wasn't actually *Given*, even though the boxes for right angles are shown in the diagram. Whatever reason you give, the fact that it is a right angle is important and must be stated!
7. BEDF is a rectangle: A parallelogram with a right angles is a rectangle.

End of Part IV

How did you do?

Questions, comments and corrections welcome.