Question

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Explanation:

Step1: Apply Pythagorean theorem for left - triangle

For a right - triangle with hypotenuse \(c\) and legs \(a\) and \(b\), \(a^{2}+b^{2}=c^{2}\). In the left - triangle, \(c = 13\) in and \(a = 12\) in. Let the unknown side be \(x\). Then \(x^{2}+12^{2}=13^{2}\).

Step2: Solve for \(x\) in the left - triangle

\[

$$\begin{align*} x^{2}&=13^{2}-12^{2}\\ x^{2}&=(13 + 12)(13 - 12)\\ x^{2}&=25\times1\\ x^{2}&=25\\ x& = 5 \end{align*}$$

\]

Step3: Apply Pythagorean theorem for right - triangle

For the right - triangle on the right with legs \(27\) and \(36\) and hypotenuse \(x\). Using \(a^{2}+b^{2}=c^{2}\), where \(a = 27\), \(b = 36\), and \(c=x\). Then \(x^{2}=27^{2}+36^{2}\).

Step4: Calculate \(x\) for the right - triangle

\[

$$\begin{align*} x^{2}&=729+1296\\ x^{2}&=2025\\ x& = 45 \end{align*}$$

\]

Answer:

The value of \(x\) for the left - triangle is \(5\) in and for the right - triangle is \(45\).