January 2024 Geometry Regents Part III

This exam was adminstered in January 2024.

January 2024 Geometry, Part III

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

32. Trish is a surveyor who was asked to estimate the distance across a pond. She stands at point C, 85 meters from point D, and locates points A and B on either side of the pond such that A, D, and B are collinear.

Trish approximates the measure of angle DCB to be 35° and the measure of angle ACD to be 75°.
Determine and state the distance across the pond, AB, to the nearest meter.

Answer:

Using the 75 degree angle, you can find the length of AD, which we'll call x. Using the 35 degree angle, you can find the length of DB, which we'll call y. Add the two together to find the length of AD.

In both cases, we have the opposite side and we need to find the adjacent side. So we need to use the tangent ratio twice.

tan 75 = x / 85
x = 85 * tan 75 = 317.224...

tan 35 = y / 85
y = 85 * tan 35 = 59.517...

AB = x + y = 317.224 + 59.517 = 376.741 = 377 meters

33. A candle in the shape of a right pyramid is modeled below. Each side of the square base measures 12 centimeters. The slant height of the pyramid measures 16 centimeters.

Determine and state the volume of the candle, to the nearest cubic centimeter
The wax used to make the candle weighs 0.032 ounce per cubic centimeter. Determine and state the weight of the candle, to the nearest ounce.

Answer:

Notice that they gave you the slant height and not the height. You need the height to find the Volume. If you take a vertical slice (cross-section) of the pyramind, you would get an isosceles triangle with a base of 12 and two legs that were 16 cm. If you draw an altitude, you will get two congruent right triangles with a base of 6 and a hypotenuse of 16. Use this information to find the height.

(6)² + b² = (16)²
36 + b² = 256
b² = 220
b = √(220) = 14.832...

Use this value to find the Volume.

V = (1/3) Area of Base * height = (1/3) * 12 * 12 * 14.832 = 711.936 = 712 cu cm.

The weight is equal to the Volume times the Density: W = (712) * (0.032) = 22.784 = 23

34. In the diagram of quadrilateral ABCD below, AB ≅ CD, and AB || CD.
Segments CE and AF are drawn to diagonal BD such that BE ≅ DF.

Prove: CE ≅ AF

Answer:

TO prove that CE is congruent to AF, you are going to have to show that triangles BEC and DFA are congruent and then use CPCTC. To show that the triangles are congruent, you can use SAS.

Your proof should look like this:

Statement	Reasons
Quadrilateral ABCD, AB ≅ CD, AB || CD, and BE ≅ DF.	Given
ABCD is a parallelogram	A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
BC ≅ AD	Opposite sides of parallelograms are congruent.
BC || AD	Opposite sides of parallelograms are parallel.
∠ CBE ≅ ∠ ADF	Alternate Interior Angles
△BCE ≅ △DAF	SAS Postulate
(AB) / (AE) = (TR) / (TE)	Corresponding sides of similar triangles are proportional
CE ≅ AF	CPCTC

End of Part III

How did you do?

Questions, comments and corrections welcome.

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