Question

22.) a=???, b= 9, c=√97

Explanation:

Assuming this is a right - triangle problem and we use 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 $b = 9$ and $c=\sqrt{97}$, and the Pythagorean theorem is $a^{2}+b^{2}=c^{2}$. Substituting the values, we get $a^{2}+9^{2}=(\sqrt{97})^{2}$.

Step 2: Simplify the equation

First, calculate $9^{2}=81$ and $(\sqrt{97})^{2}=97$. So the equation becomes $a^{2}+81 = 97$.

Step 3: Solve for $a^{2}$

Subtract 81 from both sides of the equation: $a^{2}=97 - 81$.

Step 4: Calculate the value of $a^{2}$

$97-81 = 16$, so $a^{2}=16$.

Step 5: 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. So $a=\sqrt{16}=4$.

Answer:

$a = 4$