Question

find the length of the leg x. enter the exact value, not a decimal approximation.
x =
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Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Here \(c = 15\), \(b=10\), and \(a=x\). So \(x^{2}+10^{2}=15^{2}\).

Step2: Solve for \(x^{2}\)

\(x^{2}=15^{2}-10^{2}\). Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\) or direct calculation: \(15^{2}=225\), \(10^{2}=100\). Then \(x^{2}=225-100=125\).

Step3: Find \(x\)

Take the square root of both sides. Since \(x\) represents the length of a side of a triangle (\(x>0\)), \(x=\sqrt{125}\). Simplify \(\sqrt{125}=\sqrt{25\times5}=5\sqrt{5}\).

Answer:

\(5\sqrt{5}\)