June 2017: Common Core Geometry Regents, Part 3

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.

June 2017, Geometry (Common Core), Part III

32. Triangle ABC has vertices A(-5, 2), B(-4, 7), and C(-2, 7), and triangle DEF has vertices at D(3, 2), E(2, 7), and F(0, 7). Graph and label triangle ABC and triangle DEF on the set of axes below.

Determine and state the single transformation where triangle DEF is the image of triangle ABC

Use your transformation to explain why triangle ABC = triangle DEF.

(image will be uploaded soon)
If you graph the two triangles, you will see that one is the reflection of the other. (Mirror images.) However, they are NOT reflected over the y-axis. To reflect over the y-axis, you would have to translate it first. But they want a single move.

You have to find the reflection line, which will be the vertical line halfway between points C and F, which occurs at x = -1.
So the transformation is r_{x = -1}.

Because a reflection is a rigid motion (which preserves distance and shape), DEF is congruent to ABC.

33. Given: RS and TV bisect each other at point X
TR and SV are drawn (image omitted)
Prove: TR || SV

Here’s the approach you need to take: If the lines are parallel, then the alternate interior angles along the transversals will be congruent. You can show that they are congruent by proving that the two triangles are congruent. SAS looks like the easiest approach.

Statement	Reason
1. RS and TV bisect each other at point X	Given
2. RX = XS and TX = XV	Definition of bisect
3. <TXR = <VXS	Vertical Angles are congruent
4. Triangle TXR = Triangle VXS	SAS
5. <T = <V	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
6. TR || SV	If two lines are crossed by a transversal and the alternate interior angles are congruent, then the lines are parallel.

34. A gas stations has a cylindrical fueling tank that hold the gasoline for its pumps, as modeled below. The tank holds a maximum of 20,000 gallons of gasoline and has a height of 34.5 feet. (image omitted)
A metal pole is used to measure how much gas is in the tank. To the nearest tenth of a foot, how long does the pole need to be in order to reach the bottom of the tank and still extend one foot outside the tank? Justify your answer. [I ft³ = 7.48 gallons]

Before you start, what are you looking for? The height of the stick, with is the diameter plus 1 foot. When you use the Volume formula, you will get the radius. So you need to find the radius, then double it and add 1, and then round it to the nearest tenth of a foot. Do not round to the nearest tenth in the middle of the problem. You don’t need to carry as many decimal places as I’m showing. However, I left the numbers in the calculator, so I’m showing exactly what I did. There may be minor differences in your numbers if you round that won’t affect the final answer.

First, convert gallons to cubic feet: 20,000 / 7.48 = 2673.79679
V = (pi) (r²)(h)
2673.79679 = (pi) (r²)(34.5)
r² = 2673.79679 / (34.5 * pi)
r² = 24.66944789
r = 4.96683
d = 9.93366
The stick is 10.9 feet long.

End of Part III

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.