Question

determine the missing length(s) in each triangle. write your answer as a radical in simplest form. enter the answer in the space provided. use numbers instead of words. centimeters

Explanation:

Step1: Identify the triangle type

This is a right - triangle with legs \(a = 5\) cm and \(b = 5\) cm, and hypotenuse \(c\) (the side we need to find). We use the Pythagorean theorem, which states that for a right - triangle, \(a^{2}+b^{2}=c^{2}\).

Step2: Substitute the values into the formula

Substitute \(a = 5\) and \(b = 5\) into the Pythagorean theorem:
\[

$$\begin{align*} c^{2}&=5^{2}+5^{2}\\ c^{2}&=25 + 25\\ c^{2}&=50 \end{align*}$$

\]

Step3: Solve for \(c\)

To find \(c\), we take the square root of both sides. Since \(c\) represents a length, we take the positive square root.
\[
c=\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}
\]

Answer:

\(5\sqrt{2}\)