Question

simplify the answer, rationalize the denominator if necessaryour answer to decimal form.drawwork herethe square root symbol (√), type
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Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is related to the legs \(a\) (or \(b\)) by \(c = a\sqrt{2}\). Here, \(c = 6\) and \(a=x\).

Step2: Solve for \(x\)

From \(c=a\sqrt{2}\), we can rewrite it as \(a=\frac{c}{\sqrt{2}}\). Substituting \(c = 6\) into the formula, we get \(x=\frac{6}{\sqrt{2}}\).
Rationalize the denominator: \(x=\frac{6}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\).
Convert to decimal: \(\sqrt{2}\approx1.414\), so \(x = 3\times1.414=4.242\).

Answer:

\(3\sqrt{2}\approx4.24\)