# Finding measures inside of a circle?

• Apr 2nd 2012, 10:22 PM
Hthornton
Finding measures inside of a circle?
Here are a couple of my homework problems that I need some help with. I don't know where to even start. Would someone please be kind enough as to help?

Problem 1. http://i42.tinypic.com/2ikusjn.jpg
I need to find measures of angles
(a) BAC
(b) ABD
(c) DBC
I was also suppose to find measure of arc AD and I found it to be 56, is that correct?

Problem 2. http://i43.tinypic.com/1fu0xi.jpg
Solve for x and y.
I found x = 50 but i have no idea where to start with y.

Problem 3. http://i44.tinypic.com/34zbubm.jpg
Solve for z.

I guess my biggest problem is i don't know the relation between the different inscribed angles. Any help would be greatly appreciated! Thank you very much in advance.
• Apr 3rd 2012, 06:04 AM
Soroban
Re: Finding measures inside of a circle?
Hello, Hthornton!

Quote:

Problem 1. http://i42.tinypic.com/2ikusjn.jpg
I need to find measures of angles: .(a) BAC . (b) ABD . (c) DBC

I was also suppose to find measure of arc AD
. . and I found it to be 56, is that correct? . Yes!

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

$\displaystyle \angle B\!EC = 90^o \quad\Rightarrow\quad \overarc{BC}\,= 90^o \quad\Rightarrow\quad \angle B\!AC = 45^o$

$\displaystyle \angle C\!AD = 62^o \quad\Rightarrow\quad \overarc{CD} = 124^o$

$\displaystyle \overarc{AD} \:=\:360^o - 90^o - 90^o - 124^o \:=\:56^o$

$\displaystyle \angle AB\!D = 28^o$

$\displaystyle \overarc{CD} = 124^o \quad\Rightarrow\quad \angle DBC = 62^o$
• Apr 3rd 2012, 07:49 AM
Hthornton
Re: Finding measures inside of a circle?
Thank you very much for your help, Soroban!

Hthornton
• Apr 3rd 2012, 09:12 AM
Hthornton
Re: Finding measures inside of a circle?
Would anybody be able to help me with the other two problems?
• Apr 3rd 2012, 10:51 AM
masters
Re: Finding measures inside of a circle?
Quote:

Originally Posted by Hthornton
Here are a couple of my homework problems that I need some help with. I don't know where to even start. Would someone please be kind enough as to help?

Problem 2. http://i43.tinypic.com/1fu0xi.jpg
Solve for x and y.
I found x = 50 but i have no idea where to start with y.

I guess my biggest problem is i don't know the relation between the different inscribed angles. Any help would be greatly appreciated! Thank you very much in advance.

Hi Hthornton,

Problem 2:

Angle x is equal to one-half the sum of its subtended arcs. So $\displaystyle \angle x = \frac{1}{2}(70+30) = 50$

Angle y is supplementary to angle x. They make up what is called a 'linear pair' and linear pairs are supplementary.

Problem 3:

You use the same rule as we did in problem 2. This time find the angle that is supplementary to 124 degees. That would be 180 - 124 = 56 degrees.

Now apply the same rule as in problem 2. $\displaystyle 56=\frac{1}{2}(40+z)$
• Apr 3rd 2012, 11:52 AM
Hthornton
Re: Finding measures inside of a circle?
Thank you kindly!
• Apr 3rd 2012, 04:23 PM
bjhopper
Re: Finding measures inside of a circle?
How can these calculations be performed unless AC is specified adiameter
• Apr 11th 2012, 09:54 AM
masters
Re: Finding measures inside of a circle?
Quote:

Originally Posted by bjhopper
How can these calculations be performed unless AC is specified a diameter

You wouldn't need the diameter. These are theorems used when two chords intersect inside the circle and subtend arcs on the circle. The chords do not have to be diameters.

When two chords intersect on the circle, the inscribed angle formed = 1/2 the measure of its subtended arc.

When two chords intersect inside the circle, each vertical angle formed = 1/2 the sum of the measures of their subtended arcs.
• Apr 11th 2012, 11:14 AM
bjhopper
Re: Finding measures inside of a circle?
Hello Masters,,
My comment applies only to problem 1. Tell me about that one.
• Apr 13th 2012, 03:57 AM
masters
Re: Finding measures inside of a circle?
Quote:

Originally Posted by bjhopper
Hello Masters,,
My comment applies only to problem 1. Tell me about that one.

I see what you mean. It looks like point E is the center and AC passes through E, but that's an assumption. It should have been explicitly stated.