# Geometry of a Circle

• Oct 25th 2008, 01:07 AM
BG5965
Geometry of a Circle

1) ABCDEF is a hexagon inscribed in a circle. What is x + y + z?
http://i301.photobucket.com/albums/n...agonCircle.jpg

2) In this figure, find the value of angle ABC + angle ADC and figure out if ABCD a cyclic quadrilateral. Therefore, find angle BDC
http://i301.photobucket.com/albums/n...KiteCircle.jpg

3a) In this figure, AOB is the diameter. Prove that points OAMP are concylic.
3b) Find two angles equal to angle OPA.
http://i301.photobucket.com/albums/n...leTriangle.jpg

Thanks for any help.
• Oct 25th 2008, 04:33 AM
Soroban
Hello, BG5965!

Quote:

1) $\displaystyle ABCD{E}F$ is a hexagon inscribed in a circle. .What is $\displaystyle x + y + z$ ?
http://i301.photobucket.com/albums/n...agonCircle.jpg

An inscribed angle is measured by one-half its intercepted arc.

We have: .$\displaystyle \begin{array}{ccc}x &=& \frac{1}{2}(BC + CD + DE + EF) \\ \\[-4mm] y &=& \frac{1}{2}(AB + DE + EF + FA) \\ \\[-4mm] z &=& \frac{1}{2}(FA + AB + BC + CD) \end{array}$

Add: .$\displaystyle x+y+z\;=\;\tfrac{1}{2}(2AB + 2BC + 2CD + 2DE + 2EF + 2FA)$

. . . . $\displaystyle x+y+z \;= \;AB + BC + CD + DE + EF + FA$

. . . . $\displaystyle x+y+z \;=\;360^o$

• Oct 25th 2008, 07:59 AM
Soroban
Hello again, BG5965!

I have 3(b) . . .

Quote:

3a) In this figure, $\displaystyle AOB$ is the diameter.
Prove that points OAMP are concylic.

3b) Find two angles equal to $\displaystyle \angle OPA.$

http://i301.photobucket.com/albums/n...leTriangle.jpg
I assume that $\displaystyle O$ is the center of the circle.

$\displaystyle OA = OP$ . . . both are radii.
Hence, $\displaystyle \Delta AOP$ is isosceles.
. . Therefore: .$\displaystyle \angle PAO \:=\:\angle OPA$

$\displaystyle \angle APB = 90^o$ . . . It is inscribed in a semicircle.
Right triangles $\displaystyle APB$ and $\displaystyle MOB$ share angle $\displaystyle PBA$
Then: .$\displaystyle \Delta APB \sim \Delta MOB$

. . Therefore: .$\displaystyle \angle BMO \:=\:\angle PAB \:=\:\angle OPA$