August 2018 Common Core Geometry Regents, Parts 3 and 4 (open-ended)

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

August 2018 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.

32. Given: Triangle ABC, AEC, BDE with ∠ABE = ∠CBE, and ∠ADE = ∠CDE
Prove: BDE is the perpendicular bisector of AC.

Answer:
Reason (2): "The Reflexive Property" is the reason you can say a line segment is congruent to itself
Statement (4): "<BDA = <BDC" -- ADE is congruent to CDE and statement (3) establishes that the other two angles are supplementary.
Reason (6): "CPCTC" -- Corresponding Parts of Congruent Triangles are Congruent. (Definition of Congruent Polygons)
Reason (7): If two points are each equidistant from the endpoints of a segment, then those points determine the perpendicular bisector of the segment.

Explanation of reason (7): Point B must be on the perpendicular bisector of AC. Point D must be on the perpendicular bisector of AC. Since both B and D must be on that line, then line BD must be that line that bisects AC at right angles.

This is probably not the way I would have written this proof, but it's the format we're given, and we have to fill in the blanks.

33. A homeowner is building three steps leading to a deck, as modeled by the diagram below. All three step rises, HA, FG, and DE, are congruent, and all three step runs, HG, FE, and DC, are congruent. Each step rise is perpendicular to the step run it joins. The measure of ∠CAB = 36° and m∠CBA = 90°.

If each step run is parallel to AB and has a length of 10 inches, determine and state the length of each step rise, to the nearest tenth of an inch.

Determine and state the length of AC, to the nearest inch.

Answer:
Each step run is 10 inches and there are 3 of them, so AB is 30 inches.
You have given angle A which is adjacent to AB.
For the two parts of the problem, you need to find the opposite side (and divide it by 3) and the hypotenuse.
This means using tangent and cosine, or tangent and Pythagorean Theorem.

tan 36 = x / 30
x = 30 (tan 36) = 21.796...
21.796 / 3 = 7.265. Each rise is about 7.3 inches.

cos 36 = 30 / x
x = 30 / (cos 36) = 37.08..
AC is 37 inches.

Pythagorean Theorem
30² + 21.8² = AC²
900 + 475.24 = AC²
1375.24 = AC²
AC = 37.08..., or 37 inches.

34. A bakery sells hollow chocolate spheres. The larger diameter of each sphere is 4 cm. The thickness of the chocolate of each sphere is 0.5 cm. Determine and state, to the nearest tenth of a cubic centimeter, the amount of chocolate in each hollow sphere.
The bakery packages 8 of them into a box. If the density of the chocolate is 1.308 g/cm3, determine and state, to the nearest gram, the total mass of the chocolate in the box.

Answer:
The volume of the chocolate is the Volume of the outer sphere minus the Volume of the inner sphere.
The outer has a radius of 2. The inner has a radius of 2 - 0.5 = 1.5
Use the formula V = (4/3) π r³
V = (4/3) π (2)³ - (4/3) π (1.5)³
V = 33.51 - 14.14 = 19.37 = 19.4 cm³

V = m/D, so m = VD = (19.4)(1.308) = 25.3752 each.
A package of 8 would be 25.3752 * 8 = 203 grams.

Part IV

A correct answer is worth up to 6 credits. Partial credit is available.

35. The vertices of quadrilateral MATH have coordinates M(-4,2), A(-1,–3), T(9,3), and H(6,8).
Prove that quadrilateral MATH is a parallelogram.
[The use of the set of axes is optional.]
Prove that quadrilateral MATH is a rectangle.
[The use of the set of axes below is optional.]

Answer:
Note: In my opinion, this is a fairly simple 6-point question. I would've expected this to only be worth four points, or to have another part to the question.

To prove that MATH is a parallelogram, you only need to show that the opposite sides are parallel, which means that they will have the same slope.
To prove that it is a rectangle after you have already shown that it is a parallelogram, you only need to show that any two consecutive sides are perpendicular, which means the slopes are inverse reciprocals.

You don't need to use distance formula for this problem, but you can if you want to. You could show that the opposite sides are congruent. You could show that the diagonals are congruent.

Using the graph may speed up your calculations because you can just count boxes rather than do any subtraction.

Slope MA = (-3 - 2)/(-1 - -4) = -5/3
Slope AT = (3 - -3)/9 - -1) = 6/10 = 3/5
Slope TH = (8 - 3)/(6 - 9) = 5/-3 = -5/3
Slope HM = (2 - 8)/(-4 - 6) = -6/-10 = 3/5

MATH is a parallelogram because MA is parallel to TH and AT is parallel to HM because each pair of sides has the same slope.

MATH is a rectangle because (-5/3)(3/5) = -1, so sides MA and AT are perpendicular. A parallelogram with a right angle is a rectangle.

End of Exam

How did you do?
Comments and corrections welcome. (I get many of the latter!)