Calculating Length of AD in Trapezoidal Prism
To solve for the length of \( AD \) in the trapezoidal prism, we need to use the Pythagorean Theorem in two steps because we are dealing with a three-dimensional figure.
First, we can find the length of \( BD \) by applying the Pythagorean Theorem to triangle \( BCD \), which is a right-angled triangle:
\[ BD = \sqrt{BC^2 + CD^2} \]
\[ BD = \sqrt{12^2 + 6^2} \]
\[ BD = \sqrt{144 + 36} \]
\[ BD = \sqrt{180} \]
\[ BD = 13.4164 \text{ cm} \] (approximated to four decimal places)
Now we can find the length of \( AD \) by using the Pythagorean Theorem again, this time for the right-angled triangle \( ABD \), where \( AB \) is the base, \( BD \) is the height, and \( AD \) is the hypotenuse:
\[ AD = \sqrt{AB^2 + BD^2} \]
\[ AD = \sqrt{10^2 + 13.4164^2} \]
\[ AD = \sqrt{100 + 179.9992} \]
\[ AD = \sqrt{279.9992} \]
\[ AD = 16.7426 \text{ cm} \] (approximated to four decimal places)
It seems there might be an error in the question or in the calculation, as based on the given dimensions, \( AD \) computes to 16.7426 cm rather than the 12.649 cm stated in the question. Please double-check the dimensions provided in the question and ensure any additional details or constraints need to be considered. If the dimensions given are correct, the calculated length of \( AD \) will be approximately 16.7426 cm instead of 12.649 cm.
