Circle

dhiab · #1 $Old$ November 3rd, 2009, 12:28 AM

You have :
$OA=\sqrt{50},AB=6,BC=2$
Calculate : OB

Opalg · #2 $Old$ November 3rd, 2009, 02:24 PM

Step 1. By Pythagoras' theorem, $AC = \sqrt{40}$ . Let M be the midpoint of AC. Then $AM = \sqrt{10}$ .

Step 2. OM is perpendicular to AC. By Pythagoras' theorem in the triangle OAM, $OM=\sqrt{40}$ .

Step 3. Write $\alpha$ for the angle OAB, and $\beta$ for the angle BAC. From triangle OAM, $\cos(\alpha+\beta) = 1/\sqrt5$ and $\sin(\alpha+\beta) = 2/\sqrt5$ . From triangle ABC, $\cos\beta = 3/\sqrt{10}$ and $\sin\beta = 1/\sqrt{10}$ .

Step 4. Therefore $\cos\alpha = \cos\bigl((\alpha+\beta) - \beta\bigr) = 1/\sqrt2$ (using the trig formula for the cosine of the difference of two angles).

Step 5. Now apply the cosine rule in triangle OAB to find $OB = \sqrt{26}$ .