find the value of x.\nx = ?

Explanation:

Step1: Apply the geometric - mean theorem

In a right - triangle with an altitude drawn to the hypotenuse, we have the following relationships. If (\triangle ABC) is a right - triangle with right angle at (B) and (BD\perp AC), then (AB^{2}=AD\times AC) and (BC^{2}=CD\times AC) and (BD^{2}=AD\times CD). Also, we can use the proportion (\frac{AB}{BC}=\frac{AD}{BD}=\frac{BD}{CD}). Another way is to use the fact that in right - triangle (ABC), by the Pythagorean theorem, (AC^{2}=AB^{2}+BC^{2}), and (AC=x + 4), (AB = 6), (BC = 8). So (AC=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10).

Step2: Solve for (x)

Since (AC=x + 4) and (AC = 10), we set up the equation (x+4=10). Then (x=10 - 4).

Answer:

6