Question

solve for y. simplify the answer, rationalize the denominator if necessary, but do not convert your answer to decimal form. draw show your work here hint: to add the 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 legs are equal (\(a = b\)) and the hypotenuse \(c=a\sqrt{2}\) (where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse). Here, let \(y\) be one leg and \(c = 2\).

Step2: Use the 45 - 45 - 90 triangle ratio formula

We know that \(c=y\sqrt{2}\). Substituting \(c = 2\) into the formula \(y\sqrt{2}=2\). Then solve for \(y\): \(y=\frac{2}{\sqrt{2}}\).

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{2}\): \(y=\frac{2\sqrt{2}}{\sqrt{2}\times\sqrt{2}}\). Since \(\sqrt{2}\times\sqrt{2}=2\), we have \(y = \sqrt{2}\).

Answer:

\(y=\sqrt{2}\)