Question - Calculating Length of AD in Trapezoidal Prism
Solution:
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.
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