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Retired Moderator Joined: 29 Apr 2015
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If O is the center of the circle shown above, what is the measure of x  [#permalink]

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Attachment: E89063a.png [ 7.38 KiB | Viewed 2201 times ]
If O is the center of the circle shown above, what is the measure of x?

A. 30°
B. 40°
C. 50°
D. 80°
E. 100°

Attachment: E89063c.png [ 9.79 KiB | Viewed 2189 times ]

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Re: If O is the center of the circle shown above, what is the measure of x  [#permalink]

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This is a pretty simple question that relies on the central angle and inscribed angle rules of circles. For inscribed angles (angles coming from the side of a circle), simply double them and then treat them as central angles. Here we have a central angel of 100 degrees as well as a diameter of the circle given to us. This means that the other portion of the arc on that semicircle above the diameter is 80 degrees (180-100 = 80). Because x is an inscribed angle covering that entire 80 degree portion, we use the inscribed angle rule in reverse. If the angles covers what would be 80 degrees of a central angle, then we divide that in half to reach the inscribed angle of 40 degrees, or answer choice B.

I hope this helps!
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GMAT 1: 630 Q49 V27 Re: If O is the center of the circle shown above, what is the measure of x  [#permalink]

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reto wrote:
Attachment:
The attachment E89063a.png is no longer available
If O is the center of the circle shown above, what is the measure of x?

A. 30°
B. 40°
C. 50°
D. 80°
E. 100°

answer B
Attachments gmatclub.jpg [ 25.39 KiB | Viewed 2112 times ]

Manager  B
Joined: 15 Nov 2017
Posts: 54
Re: If O is the center of the circle shown above, what is the measure of x  [#permalink]

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Could someone provide more insight about the rules related to this problem? I am having trouble correctly solving this one. Thank you!

reto wrote:
Attachment:
E89063a.png
If O is the center of the circle shown above, what is the measure of x?

A. 30°
B. 40°
C. 50°
D. 80°
E. 100°
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3251
If O is the center of the circle shown above, what is the measure of x  [#permalink]

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1
KHow wrote:
Could someone provide more insight about the rules related to this problem? I am having trouble correctly solving this one. Thank you!

reto wrote:
Attachment:
The attachment E89063a.png is no longer available
If O is the center of the circle shown above, what is the measure of x?

A. 30°
B. 40°
C. 50°
D. 80°
E. 100°

Attachment: inscribed angles 2019.1.17.png [ 39.99 KiB | Viewed 527 times ]

KHow , sure.
One central angle and one inscribed angle intercept the same arc, in my diagram, arc AB
Central angle measure yields arc measure
Arc measure yields inscribed angle measure $$= x$$

A central angle is formed by two radii, OA and OB with its vertex at center O

• degree measure of central angle ∠ AOB
Straight line = 180°
Two angles that lie on a straight line sum to 180°
100° + ∠AOB = 180°
∠ AOB = 80°

1) ∠ AOB is "subtended" by arc AB, or ∠ AOB intercepts arc AB

2) degree measure of intercepted arc AB ?

The radii of sector AOB intercept the circumference of the circle and create arc AB.

We cam also say that ∠ AOB "subtends" arc AB

Rule: The length of an arc subtended by a central angle has the same degree measure as the central angle

∠ AOB = 80°

Because ∠ AOB is a central angle, arc AB also = 80°

3) from degree measure of arc AB, find inscribed ∠ ACB measure = $$x$$

An inscribed angle is formed by two intersecting chords, AC and BC, whose vertex lies on the circumference of the circle

4) An inscribed angle has a degree measure that is $$\frac{1}{2}$$ that of the arc it intercepts

Rule: the degree measure of an inscribed angle is half that of the arc that the angle intercepts

5) arc AB = 80°

6) x =?
∠ ACB = $$x = \frac{1}{2}$$ of arc AB

$$x = \frac{1}{2} * 80°$$

$$x = 40°$$

Answer B

Hope that helps.

This site, here, and this site, here discuss the relationship between and among inscribed angles, central angles, and intercepted arcs
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Manager  B
Joined: 15 Nov 2017
Posts: 54
Re: If O is the center of the circle shown above, what is the measure of x  [#permalink]

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1
This is VERY helpful. Finally get it. Thanks a lot! generis wrote:
KHow wrote:
Could someone provide more insight about the rules related to this problem? I am having trouble correctly solving this one. Thank you!

reto wrote:
Attachment:
E89063a.png
If O is the center of the circle shown above, what is the measure of x?

A. 30°
B. 40°
C. 50°
D. 80°
E. 100°

Attachment:
inscribed angles 2019.1.17.png

KHow , sure.
One central angle and one inscribed angle intercept the same arc, in my diagram, arc AB
Central angle measure yields arc measure
Arc measure yields inscribed angle measure $$= x$$

A central angle is formed by two radii, OA and OB with its vertex at center O

• degree measure of central angle ∠ AOB
Straight line = 180°
Two angles that lie on a straight line sum to 180°
100° + ∠AOB = 180°
∠ AOB = 80°

1) ∠ AOB is "subtended" by arc AB, or ∠ AOB intercepts arc AB

2) degree measure of intercepted arc AB ?

The radii of sector AOB intercept the circumference of the circle and create arc AB.

We cam also say that ∠ AOB "subtends" arc AB

Rule: The length of an arc subtended by a central angle has the same degree measure as the central angle

∠ AOB = 80°

Because ∠ AOB is a central angle, arc AB also = 80°

3) from degree measure of arc AB, find inscribed ∠ ACB measure = $$x$$

An inscribed angle is formed by two intersecting chords, AC and BC, whose vertex lies on the circumference of the circle

4) An inscribed angle has a degree measure that is $$\frac{1}{2}$$ that of the arc it intercepts

Rule: the degree measure of an inscribed angle is half that of the arc that the angle intercepts

5) arc AB = 80°

6) x =?
∠ ACB = $$x = \frac{1}{2}$$ of arc AB

$$x = \frac{1}{2} * 80°$$

$$x = 40°$$

Answer B

Hope that helps.

This site, here, and this site, here discuss the relationship between and among inscribed angles, central angles, and intercepted arcs Re: If O is the center of the circle shown above, what is the measure of x   [#permalink] 18 Jan 2019, 11:08
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# If O is the center of the circle shown above, what is the measure of x

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