Mathematical: Geometric Principles Flashcards

A rectangular garden is 12 feet long and 8 feet wide. The owner wants to install a fence around the entire perimeter and cover the garden with a layer of topsoil. Which statement accurately describes the amount of fencing and topsoil needed?

Answer: 40 feet of fencing and 96 square feet of topsoil are needed.

The amount of fencing required is the perimeter of the garden. The formula for the perimeter of a rectangle is P = 2(length + width). So, P = 2(12 ft + 8 ft) = 2(20 ft) = 40 feet. The amount of topsoil needed is the area of the garden. The formula for the area of a rectangle is A = length × width. So, A = 12 ft × 8 ft = 96 square feet.

A 13-foot ladder is leaning against a vertical wall. The bottom of the ladder is 5 feet from the base of the wall. How high up the wall does the ladder reach?

Answer: 12 feet

The ladder, wall, and ground form a right triangle, with the ladder as the hypotenuse. The Pythagorean theorem, a² + b² = c², can be used to solve this. Here, c = 13 and b = 5. So, a² + 5² = 13², which is a² + 25 = 169. Subtracting 25 from both sides gives a² = 144. Taking the square root of 144 gives a = 12. The ladder reaches 12 feet up the wall.

Two angles are supplementary. If the measure of the first angle is 115°, what is the measure of the second angle?

Answer: 65°

Supplementary angles are two angles whose measures add up to 180 degrees. To find the measure of the second angle, subtract the known angle from 180: 180° - 115° = 65°.

A point is located at (-3, 5) on a coordinate plane. If this point is reflected across the y-axis, what will its new coordinates be?

Answer: (3, 5)

When a point is reflected across the y-axis, the y-coordinate remains the same, while the x-coordinate becomes its opposite. The opposite of -3 is 3. Therefore, the new coordinates are (3, 5).

A shipping company uses a box shaped like a rectangular prism with a length of 10 inches, a width of 6 inches, and a height of 4 inches. What is the total volume of the box?

Answer: 240 cubic inches

The formula for the volume of a rectangular prism is V = length × width × height. Plugging in the given dimensions: V = 10 in × 6 in × 4 in = 240 cubic inches.

Which of the following statements describes a property that is ALWAYS true for any rhombus?

Answer: All four sides are equal in length.

By definition, a rhombus is a quadrilateral where all four sides have the same length. While some rhombuses (squares) have right angles and equal diagonals, this is not true for all rhombuses. A rhombus is a parallelogram, so both pairs of opposite sides are parallel.