Saturday, February 15, 2020

January 2020 Geometry Regents, Part 2

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

January 2020 Geometry, Part II

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

25. In the diagram below, right triangle PQR is transformed by a sequence of rigid motions that maps it onto right triangle NML.

Write a set of three congruency statements that would show ASA congruency for these triangles.

Pick two pairs of corresponding angles and a pair of corresponding sides.
Note: Use the Congruency symbol, not the Equal sign. (I don't have the option on this blog.)
<Q = <M ; <R = <L ; QR = ML ;

26. Diego needs to install a support beam to hold up his new birdhouse, as modeled below. The base of the birdhouse is 24 1/2 inches long. The support beam will form an angle of 38° with the vertical post. Determine and state the approximate length of the support beam, x, to the nearest inch.

We don't care about the birdhouse. We only care about the right triangle under it.
We know the side opposite the angle and the hypotenuse, so we need to use Sine.

Sin 38° = 24.5 / x
x = 24.5 / Sin 38° = 39.79 = 40 inches.

27. A rectangular tabletop will be made of maple wood that weighs 43 pounds per cubic foot. The tabletop will have a length of eight feet, a width of three feet, and a thickness of one inch.
Determine and state the weight of the tabletop, in pounds.

To find the weight, you first need to find the Volume in cubic feet, and then multiply it by the weight of the wood. You have to convert the inch into 1/12 feet.

V = l * w * h = (8) (3) (1/2) = 2
W = 43 * 2 = 86 pounds

28. In the diagram below of circle O, secant ABC and tangent AD are drawn.

If CA = 12.5 and CB = 4.5, determine and state the length of DA.

Tangent-Secant Rule: (AD)2 = (AB)(AC)
(AD)2 = (12.5 - 4.5)(12.5)
(AD)2 = 100
DA = 10

29. Given ___ MT below, use a compass and straightedge to construct a 45° angle whose vertex is at point M. [Leave all construction marks.]

The easiest way I would think to do this is to create a 90-45-45 triangle, which M as one of the vertices. You construct a perpendicular bisector of MT, and then draw an arc from the right angle that has a length equal to the base of the triangle.

Step one: From M, make arcs above and below the line that go at least halfway across.
Then, without changing the size of the compass, go to T and repeat the process. You should have marked off two Xs.
With a straightedge, draw a line between the center of those two Xs. That is a perpendicular bisector.
Let's give the right angle a name, call it "R". You don't have to do this, but it will help me explain the following step.

Step two: From R, open the compass to point M. Draw an arc up to the vertical line. Call that point "S". With a straightedge, draw MS. Label angle M 45°. (Note that angle MSR is also 45°.)


30. In triangle XYZ shown below, medians XE, YF, and ZD intersect at C.
If CE = 5, YF = 21, and XZ = 15, determine and state the perimeter of triangle CFX.

The medians cut the sides of XYZ in half. The medians themselves intersect at the centroid which divides the median into a ratio of 2:1.

If CE = 5, that means that CX = 10 (and EX = 15).
If YF = 21, that means that CF = 7 (and YC = 14).
If XZ = 15, that FX = 7.5.
So the perimeter of CFX ix 10 + 7 + 7.5 = 24.5.

31. Determine and state an equation of the line perpendicular to the line 5x - 4y = 10 and passing through the point (5,12).

Short answer: Since it is in Standard form, switch the coefficients and change the sign to get a perpendicular line.
4x + 5y = C
4(5) + 5(12) = 20 + 60 = 80
4x + 5y = 80

Second answer: Use Point-Slope form.
The slope of 5x - 4x = 10 is -(5/-4) = 5/4
The perpendicular slope is the inverse reciprocal -4/5.
y - 12 = -4/5(x - 5)

Long way: put everything in slope-intercept form and slog it out:
5x - 4y = 10
-4y = -5x + 10
y = 5/4x - 10/4
Perpendicular line: y = -4/5 x + b
12 = -4/5 (5) + b
12 = -4 + b
16 = b
y = -4/5 x + 16

End of Part II

How did you do?

Questions, comments and corrections welcome.

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