Question

1. $a^{2}+b^{2}=c^{2}$ $12^{2}+x^{2}=$ 6. $x = 15$

Explanation:

Step1: Apply Pythagorean theorem

$$a^{2}+b^{2}=c^{2}$$
In the right - triangle, \(a = 12\), \(b=9\), and \(c=x\). So \(12^{2}+9^{2}=x^{2}\).

Step2: Calculate the squares

\(12^{2}=144\), \(9^{2}=81\). Then \(x^{2}=144 + 81\).

Step3: Sum the values

\(144+81=225\), so \(x^{2}=225\).

Step4: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{225}\). Since \(x>0\) (length of a side of a triangle), \(x = 15\).

Answer:

\(x = 15\)