right triangle \( A B C \), angle \( C \) is a right angle, \( A B=17 \), and \( B C=15 \). What is the ngth of \( A C \) ? Show using the Right-Triangle Theorem formula and label \( \mathrm{a}, \mathrm{b}, \mathrm{c} \).
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To find the length of side \( AC \) in right triangle \( ABC \), we can apply the Pythagorean theorem, which states that \( a^2 + b^2 = c^2 \). Here, \( c \) represents the hypotenuse \( AB \) (which is 17), and the legs are \( BC \) (which is 15) and \( AC \). Let's label the sides: - \( a = BC = 15 \) - \( b = AC \) (unknown) - \( c = AB = 17 \) Now we substitute into the Pythagorean theorem: \[ 15^2 + AC^2 = 17^2 \] \[ 225 + AC^2 = 289 \] Subtracting 225 from both sides gives: \[ AC^2 = 64 \] Taking the square root, we find: \[ AC = 8 \] So, the length of \( AC \) is 8 units!
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