Question

23.) a=???, b=5, c=√106

Explanation:

Assuming this is a right - triangle problem and we are using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse and \(a,b\) are the legs of the right triangle).

Step 1: Substitute the known values into the Pythagorean theorem

We know that \(b = 5\) and \(c=\sqrt{106}\). Substituting these values into the formula \(a^{2}+b^{2}=c^{2}\), we get:
\(a^{2}+5^{2}=(\sqrt{106})^{2}\)

Step 2: Simplify the equation

First, calculate the squares:
\(5^{2}=25\) and \((\sqrt{106})^{2}=106\)
So the equation becomes:
\(a^{2}+25 = 106\)

Step 3: Solve for \(a^{2}\)

Subtract 25 from both sides of the equation:
\(a^{2}=106 - 25\)
\(a^{2}=81\)

Step 4: Solve for \(a\)

Take the square root of both sides. Since \(a\) represents the length of a side of a triangle, we take the positive square root:
\(a=\sqrt{81}=9\)

Answer:

\(a = 9\)