Question - Calculating hypotenuse length using geometric mean relationships
mathright triangle altitudecalculating hypotenuse lengthgeometric mean relationshipstriangle geometry
Solution:
In the given right triangle $$ ABC $$, with altitude $$ BD $$ drawn to the hypotenuse $$ AC $$, and $$ BD = 6 $$ and $$ DC = 12 $$, we can find the length of $$ AC $$ by using the geometric mean relationships that exist in a right triangle with an altitude drawn to its hypotenuse.The length of $$ AC $$ is the sum of $$ AD $$ and $$ DC $$. We know $$ DC = 12 $$, but we need to find $$ AD $$ before we can find $$ AC $$.The altitude $$ BD $$ is the geometric mean of the two segments of the hypotenuse. So we have $$ BD^2 = AD \cdot DC $$.Given that $$ BD = 6 $$ and $$ DC = 12 $$, we plug these values into the equation:$$ 6^2 = AD \cdot 12 $$$$ 36 = AD \cdot 12 $$Now, solve for $$ AD $$:$$ AD = \frac{36}{12} = 3 $$Now that we know $$ AD $$ is 3 and $$ DC $$ is 12, we can find the length of $$ AC $$ by adding these two lengths:$$ AC = AD + DC = 3 + 12 = 15 $$So, the length of the hypotenuse $$ AC $$ is 15 units.
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