Question

1. in right △abc, (overline{cd}) is the altitude drawn to hypotenuse (overline{ab}). if (cd = 6), (ad = 3), and (db = 5x - 3), find (x).

Explanation:

Step1: Use geometric mean theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments of the hypotenuse. So, \(CD^{2}=AD\times DB\).

Step2: Substitute the given values

Given \(CD = 6\), \(AD = 3\), and \(DB=5x - 3\). Substitute into the formula: \(6^{2}=3\times(5x - 3)\).

Step3: Simplify the equation

First, calculate \(6^{2}=36\). Then the equation becomes \(36 = 15x-9\).

Step4: Solve for \(x\)

Add \(9\) to both sides: \(36 + 9=15x\), so \(45 = 15x\). Divide both sides by \(15\): \(x=\frac{45}{15}\).

Answer:

\(x = 3\)