central angle theorem

**MATNTRNG** · September 14th 2010, 09:12 PM

Use the angles sum of a triangle theorem and the linear pair theorem to prove the following result:

If triangle ABC is a right triangle with right angle at C and O is the midpoint of AB, then angle BOC = 2 * angle BAC.

**Soroban** · September 15th 2010, 05:22 AM

Hello, MATNTRNG!

$\text{Use the angles sum of a triangle theorem and the linear pair theorem}$
$\text{to prove the following result:}$

$\text{If }\Delta ABC\text{ is a right triangle with }C = 90^o\text{ and }O\text{ is the midpoint of }AB,$
$\text{then: }\:\angle BOC \:=\: 2\!\cdot\!\angle BAC$

I have my own proof of this . . .

Code:

    A o
      * *   r
      * @ *
      *     *   O
      *       o
      *     * 2@*   r
      * @ * r     *
      * *           *
    C o   *   *   *   o B

Right triangle $ABC$ can be inscribed in a semicircle.
. . $O$ is the center of the circle
. . and $OA, OB, OC$ are radii $\,r$ of the circle.

Hence, $\Delta AOC$ is isosceles and $\angle OAC = \angle ACO = \theta$

$\angle BOC$ is an exterior angle of $\Delta AOC$ .
. . Hence: . $\angle BOC \:=\:\angle OAC + \angle ACO \;=\;\theta + \theta \;=\;2\theta$

Therefore: . $\angle BOC \;=\;2\!\cdot\!\angle OAC \;=\;2\!\cdot\!\angle BAC$

Thread: central angle theorem

LinkBack

Thread Tools

Display

central angle theorem

Similar Math Help Forum Discussions

Finding central angle after slicing cone

Central angle of circle

Prove a central angle is twice the measure of an inscribed angle

Central angle help

Finding the radius with just the chord length and central angle :)

Search Tags