Geometry Problems of the Day (Geometry Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 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. A drone is used to measure the size of a brush fire on the ground. Segment AB represents the width of the fire, as shown below. The drone calculates the distance to point B to be 1076 feet at an angle of depression of 25°. At the same point, the drone calculates the distance to point A to be 774 feet at an angle of depression of 36°.

Determine and state the width of the fire, AB, to the nearest foot.

Answer:

Call the right angle C. To find the length of AB, you need to find the lengths of AC and BC and then subtract.

Because the two horizontal lines shown in the picture are parallel (all horizontal lines have slope = 0 and are therefore parallel), then we know that Angle A has a measure of 36° and Angle B has a measure of 25°.

In both cases, we have the hypotenuse and we are looking for the adjacent side of the right triangle. (The opposite side is the height. We don't know it, and we are not looking for it.) This means we must use cosine. Twice.

Cos 25 = x / 1076
x = 1076 cos 25 = 975.1871...

Cos 36 = x / 774
x = 774 cos 36 = 626.1791...

975.1871 - 626.1791 = 349.008

The length of AB is 349 feet.

33. 3 Quadrilateral ABCD has vertices with coordinates A(-3,6), B(6,3), C(6,-2), and D(-6,2). Joe defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Joe’s definition to prove ABCD is an isosceles trapezoid. [The use of the set of axes below is optional.]

Answer:

You don't have to graph the figure but it will make the problem easier to visualize. Also, you must show that this is an isosceles trapezoid using Joe's method or you will not receive full credit.

First, show that the figure is a trapezoid by showing that there is only one pair of parallel sides with the same slope. The slope of AB is -1, and the slope of CD is -1, so AB || CD. The slope of BC is undefined (a vertical line), and the slope of DA is 4/3, so BC is NOT parallel to DA. Therefore, ABCD is a trapezoid.

Second, in a trapezoid, the diagonals are congruent. so use the Distance Formula (or Pythagorean Theorem) to find the lengths of diagonals AC and BD.

AC = &sqrt;( (-3 - 6)² + (6 - -2)²) = &sqrt;(81 + 64) = &sqrt;(145)

BD = &sqrt;( (-6 - 6)² + (3 - 2)²) = &sqrt;(144 + 1) = &sqrt;(145)

Since AC ≅ BD the trapezoid is isosceles.

If you found that AD is congruent to BC, then you would not get full credit for this problem.

34. Ali made six solid spherical decorations out of modeling clay. Each decoration has a radius of 2.5 inches. The weight of clay is 68 pounds per cubic foot.

Determine and state, to the nearest pound, the total weight of the six decorations.

Answer:

You are given the weight per cubic foot. You need to find the number of cubic feet total in all six decorations. However, you are given measurements in inches, so you'll need to divide by 12³.

V = 4/3 π r³ = 4/3 π (2.5)³ = 65.4498 ...

Six decorations have a volume of 6 * 65.4498 = 392.6990 cubic inches or 392.6990/12³ = 0.227 cubic feet

The weight is 0.227 * 68 = 15.436 or 15 pounds.

End of Part III

How did you do?

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