Question

solve for c. do not convert your answer to decimal form and rationalize the denominator if necessary. 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 right - angled isosceles triangle (since one angle is \(90^{\circ}\) and another is \(45^{\circ}\), the third angle is also \(45^{\circ}\)). In a \(45 - 45-90\) triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Let the legs be of length \(a = b=20\) (the two non - hypotenuse sides). The formula for the hypotenuse \(c\) of a right - angled triangle is \(c=\sqrt{a^{2}+b^{2}}\) (Pythagorean theorem).

Step2: Substitute values into the formula

Substitute \(a = 20\) and \(b = 20\) into \(c=\sqrt{a^{2}+b^{2}}\). So \(c=\sqrt{20^{2}+20^{2}}=\sqrt{400 + 400}=\sqrt{800}\).

Step3: Simplify the square root

We can simplify \(\sqrt{800}\) as \(\sqrt{100\times8}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 100\), \(b = 8\)), we get \(\sqrt{100\times8}=\sqrt{100}\times\sqrt{8}=10\times\sqrt{4\times2}\). Again, using \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 4\), \(b = 2\)), we have \(10\times\sqrt{4}\times\sqrt{2}=10\times2\times\sqrt{2}=20\sqrt{2}\).

Answer:

\(20\sqrt{2}\)