Question

in circle o shown below, diameter \\(\overline{db}\\) is perpendicular to chord \\(\overline{ac}\\) at e.

if \\(db = 34\\), \\(ac = 30\\), and \\(de > be\\), what is the length of \\(\overline{be}\\)?
\\(\bigcirc\\) 16
\\(\bigcirc\\) 25
\\(\bigcirc\\) 8
\\(\bigcirc\\) 9

Explanation:

Step1: Find radius of circle

The diameter $DB = 34$, so the radius $OA = OB = \frac{34}{2} = 17$.

Step2: Find length of $AE$

Since diameter $DB \perp$ chord $AC$, it bisects $AC$. So $AE = \frac{AC}{2} = \frac{30}{2} = 15$.

Step3: Calculate length of $OE$

Use Pythagorean theorem in $\triangle OAE$:
$$OE = \sqrt{OA^2 - AE^2}$$
$$OE = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8$$

Step4: Calculate length of $BE$

Since $OB = 17$ and $OE = 8$, $BE = OB - OE = 17 - 8 = 8$.

Answer:

8