NJSLA Geometric Principles and Measurement Questions and Answers — Questions and Answers
Question 1: A cylindrical water tank has a radius of 5 feet and a height of 12 feet. A cone with the exact same radius and height is placed inside the tank. If the tank was initially full of water, which of the following represents the volume of water remaining in the tank after the cone is placed inside?
- 100π cubic feet
- 300π cubic feet
- 200π cubic feet (Correct answer)
- 50π cubic feet
Correct answer: 200π cubic feet
First, calculate the volume of the cylinder using the formula V = πr²h. V_cylinder = π(5²)(12) = 300π cubic feet. Next, calculate the volume of the cone using the formula V = (1/3)πr²h. V_cone = (1/3)π(5²)(12) = 100π cubic feet. To find the remaining volume, subtract the volume of the cone from the volume of the cylinder: 300π - 100π = 200π cubic feet.
Question 2: A rectangular park is 120 meters long and 50 meters wide. A straight diagonal path connects two opposite corners of the park. What is the length of the path?
- 170 meters
- 130 meters (Correct answer)
- 110 meters
- 150 meters
Correct answer: 130 meters
The length and width of the rectangle form the two legs of a right triangle, and the diagonal path is the hypotenuse. Using the Pythagorean theorem (a² + b² = c²): 50² + 120² = c². This becomes 2500 + 14400 = c², which simplifies to 16900 = c². Taking the square root of both sides gives c = 130 meters.
Question 3: In the coordinate plane, Triangle ABC has vertices at A(2, 3), B(5, 3), and C(2, 7). Which of the following transformations will result in a triangle with vertices at A'(-2, -3), B'(-5, -3), and C'(-2, -7)?
- A reflection across the y-axis
- A translation 4 units to the left
- A reflection across the x-axis
- A 180-degree rotation about the origin (Correct answer)
Correct answer: A 180-degree rotation about the origin
A 180-degree rotation about the origin follows the rule (x, y) -> (-x, -y). Applying this rule to each vertex: A(2, 3) becomes A'(-2, -3), B(5, 3) becomes B'(-5, -3), and C(2, 7) becomes C'(-2, -7). These are the coordinates of the transformed triangle.
Question 4: Two parallel lines are intersected by a transversal line. If the measure of one of the alternate interior angles is 55°, what is the measure of its consecutive interior angle?
- 55°
- 35°
- 125° (Correct answer)
- 90°
Correct answer: 125°
When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, meaning their sum is 180°. If one angle is 55°, its consecutive interior angle is 180° - 55° = 125°. Alternate interior angles are congruent, but the question asks for the consecutive interior angle.
Question 5: A contractor is hired to install a new floor in a rectangular room that measures 15 feet by 20 feet. There is a square-shaped support column in the room with a side length of 2 feet. What is the total area of flooring the contractor will need?
- 300 square feet
- 304 square feet
- 296 square feet (Correct answer)
- 260 square feet
Correct answer: 296 square feet
First, calculate the total area of the room by multiplying its length and width: Area_room = 15 ft * 20 ft = 300 sq ft. Next, calculate the area of the support column: Area_column = 2 ft * 2 ft = 4 sq ft. Since the flooring will not be placed under the column, subtract the column's area from the room's area: 300 sq ft - 4 sq ft = 296 sq ft.
Question 6: A landscaper is designing a circular garden with a diameter of 22 feet. They plan to build a low fence around the entire border of the garden. Which of the following is the closest to the length of fencing needed?
- 380.1 feet
- 34.6 feet
- 138.2 feet
- 69.1 feet (Correct answer)
Correct answer: 69.1 feet
The length of the fence corresponds to the circumference of the circle. The formula for circumference is C = πd, where d is the diameter. Using the given diameter of 22 feet: C = π * 22. Using π ≈ 3.14159, the circumference is approximately 69.115 feet. The closest answer is 69.1 feet.
