Question

find the value of x.

x=

tell whether the side lengths form a pythagorean triple.
yes
no

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Here \(a = 6\), \(c=10\), and \(b=x\). So \(x^{2}+6^{2}=10^{2}\).

Step2: Simplify the equation

\(x^{2}+36 = 100\). Then \(x^{2}=100 - 36\).

Step3: Calculate \(x^{2}\) and \(x\)

\(x^{2}=64\), so \(x=\sqrt{64}\). Since \(x>0\) (length of a side), \(x = 8\).

Step4: Check Pythagorean triple

A Pythagorean triple consists of three positive integers \(a,b,c\) such that \(a^{2}+b^{2}=c^{2}\). Here \(6^{2}+8^{2}=36 + 64=100=10^{2}\), and \(6,8,10\) are all integers.

Answer:

\(x = 8\), yes.