Question

which of the following are ways in which the pythagorean theorem can be useful in solving a practical problem? select all that apply. a. the theorem can be used to calculate fractional degrees of measurement. b. the theorem can be used to find the optimal area of the square, fenced yard on a limited budget for fence material. c. the theorem can help you find the length of a third, unknown distance in the a right triangle. d. the theorem can be used to find the rate at which something is increasing or decreasing when the situation is modeled as a right triangle. e. the theorem can be useful in calculating the angular size of an object such as the moon. f. the theorem can be used to find the optimal size of a sqaure pyramid container.

Explanation:

The Pythagorean theorem ($a^{2}+b^{2}=c^{2}$ for a right - triangle with legs $a$ and $b$ and hypotenuse $c$) is used to find the length of a side of a right - triangle when the other two sides are known. It can also be applied in real - world situations modeled as right - triangles to find rates of change or optimal sizes related to right - triangular shapes.

• Option A: The Pythagorean theorem is not for calculating fractional degrees of measurement.
• Option B: Finding the optimal area of a square fenced yard on a budget is related to perimeter and area concepts, not the Pythagorean theorem.
• Option C: This is a direct use of the Pythagorean theorem to find an unknown side of a right - triangle.
• Option D: In related rates problems modeled as right - triangles, the Pythagorean theorem can be used to find rates of change.
• Option E: Calculating the angular size of an object like the moon uses trigonometric concepts related to angles and distances, not the Pythagorean theorem.
• Option F: For a square pyramid container, if there are right - triangular relationships within its structure (e.g., slant height, height, and base - related lengths), the Pythagorean theorem can be used to find optimal sizes.

Answer:

C. The theorem can help you find the length of a third, unknown distance in the a right triangle.
D. The theorem can be used to find the rate at which something is increasing or decreasing when the situation is modeled as a right triangle.
F. The theorem can be used to find the optimal size of a square pyramid container.