Question

find the missing side length in the triangle shown. express your answer as a completely simplified square root. options: z = 25√2, z = 2√5, z = 5√2, z = √10

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle with two legs of length 5. We can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 5\), \(b = 5\) and \(c=z\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 5\) and \(b = 5\) into the formula: \(z^{2}=5^{2}+5^{2}\). Calculate \(5^{2}=25\), so \(z^{2}=25 + 25=50\).

Step3: Solve for \(z\)

Take the square root of both sides: \(z=\sqrt{50}\). Simplify \(\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\).

Answer:

\(z = 5\sqrt{2}\) (corresponding to the option with \(z = 5\sqrt{2}\))