Question

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Explanation:

Step1: Apply Pythagorean theorem

For the first right - triangle with sides 5, x, and 10. The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where c is the hypotenuse. Here, \(5^{2}+x^{2}=10^{2}\), so \(x^{2}=10^{2}-5^{2}\).
\[x^{2}=100 - 25=75\]
\[x=\sqrt{75}=5\sqrt{3}\]

Step2: Apply Pythagorean theorem for the second triangle

For the second right - triangle with sides 11, x, and 13. Using the Pythagorean theorem \(11^{2}+x^{2}=13^{2}\), then \(x^{2}=13^{2}-11^{2}\).
\[x^{2}=169 - 121 = 48\]
\[x=\sqrt{48}=4\sqrt{3}\]

Answer:

For the first triangle: \(x = 5\sqrt{3}\), for the second triangle: \(x=4\sqrt{3}\)