Question

use the pythagorean theorem to find x and y in each of the following. complete parts (a) through (f) below.
c. x = 16 (simplify your answer. type an exact answer, using radicals as needed.)
d. x = \\( \frac { \sqrt { 3 } s } { 2 } \\) (simplify your answer. type an exact answer, using radicals as needed. type an expression using s as the variable. use integers or fractions for any numbers in the expression.)
e. x = \\( \square \\) and y = \\( \square \\) (simplify your answers. type exact answers, using radicals as needed.)

Explanation:

Step1: Find \( y \) (diagonal of the base)

The base is a square with side length \( 2 \). Using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \) for the right - triangle formed by the sides of the base (\( a = b=2 \)):
\( y^{2}=2^{2}+2^{2} \)
\( y^{2}=4 + 4 \)
\( y^{2}=8 \)
\( y=\sqrt{8}=2\sqrt{2} \)

Step2: Find \( x \) (space - diagonal)

Now, consider the right - triangle formed by \( y = 2\sqrt{2} \) (one leg) and the height of the prism \( h = 2 \) (another leg). Using the Pythagorean theorem \( x^{2}=y^{2}+2^{2} \).
Since \( y^{2}=8 \), then \( x^{2}=8 + 4 \)
\( x^{2}=12 \)
\( x=\sqrt{12}=2\sqrt{3} \)

Answer:

\( x = 2\sqrt{3}\) and \( y=2\sqrt{2}\)