reviewing the pythagorean theorem $a^2 + b^2 = c^2$
calculate the value of x?
image of a right triangle with legs 5 and 12, hypotenuse x
now you are ready to investigate and look for a pattern!! use the pythagorean theorem to calculate the length of the diagonals of the squares. simplify your radicals as we did in the warm up!
1) $x = ???$ (show your work below!!)
image of a square with side 5, diagonal x, and 45° angles at the diagonals endpoints

Question

Accepted Answer

# Explanation:
## Step1: Identify the triangle type
The first triangle is a right - triangle with legs $a = 5$ and $b=12$, and hypotenuse $x$. The second figure is a square with side length $5$, and the diagonal $x$ forms a right - triangle with two sides of the square (both equal to $5$).

## Step2: Calculate $x$ for the right - triangle (first part)
For a right - triangle, we use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse. Here, $a = 5$, $b = 12$, so:
$$
\begin{align*}
x^{2}&=5^{2}+12^{2}\
x^{2}&=25 + 144\
x^{2}&=169\
x&=\sqrt{169}\
x& = 13
\end{align*}
$$

## Step3: Calculate $x$ for the square's diagonal (second part)
In the square, the two legs of the right - triangle formed by the diagonal are both $a=b = 5$. Using the Pythagorean theorem $x^{2}=5^{2}+5^{2}$:
$$
\begin{align*}
x^{2}&=25+25\
x^{2}&=50\
x&=\sqrt{50}\
x&=\sqrt{25\times2}\
x& = 5\sqrt{2}
\end{align*}
$$

# Answer:
For the first right - triangle, $x = 13$; for the square's diagonal, $x=5\sqrt{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the triangle type

Step2: Calculate \(x\) for the right - triangle (first part)

Step3: Calculate \(x\) for the square's diagonal (second part)

Answer: