Question

what is the pythagorean theorem used for in three - dimensional space?
a. finding the area of a triangle
b. calculating straight - line distances between points
c. measuring the volume of a prism
d. determining the circumference of a circle

a triangular prism has a right triangular base with legs of 9 m and 12 m. if the height of the prism is 15 m, what is the space diagonal?
a. 20 m
b. 24.85 m
c. 22.36 m
d. 21.21 m

Explanation:

Step1: Recall Pythagorean - Theorem use

The Pythagorean Theorem in 3 - D space is used for calculating straight - line distances between points. It is an extension of finding distances in 2 - D right - triangles to 3 - D space.

Step2: Solve for space diagonal of triangular prism

First, find the hypotenuse of the base right - triangle using the Pythagorean Theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 9$ m and $b = 12$ m. So, $c=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15$ m.
Then, find the space diagonal $d$ of the prism using the 3 - D Pythagorean Theorem $d=\sqrt{c^{2}+h^{2}}$, where $c = 15$ m (hypotenuse of base) and $h = 15$ m (height of prism). So, $d=\sqrt{15^{2}+15^{2}}=\sqrt{225+225}=\sqrt{450}\approx21.21$ m.

Answer:

1. b. Calculating straight - line distances between points
2. d. 21.21 m