Question

solve for z. 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 right - angled isosceles triangle (since one angle is \(45^{\circ}\) and it's a right - triangle, the other non - right angle is also \(45^{\circ}\)). For a right - angled isosceles triangle with hypotenuse \(c\) and legs \(a\) and \(b\) (here \(a = b=z\)), the Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). Substituting \(a = b = z\) and \(c = 20\), we get \(z^{2}+z^{2}=20^{2}\).

Step2: Simplify the equation

Combine like terms: \(2z^{2}=400\). Then divide both sides by 2: \(z^{2}=\frac{400}{2}=200\).

Step3: Solve for \(z\)

Take the square root of both sides: \(z=\sqrt{200}\). Simplify \(\sqrt{200}=\sqrt{100\times2}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a = 100\), \(b = 2\)), we have \(z = 10\sqrt{2}\).

Answer:

\(z = 10\sqrt{2}\)