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The length of minor arc AB is twice the length of minor arc

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enigma123
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The length of minor arc AB is twice the length of minor arc [#permalink] New post  11 Feb 2012, 14:55
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The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120

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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  11 Feb 2012, 15:18
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The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120


(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Answer: B.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.

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Arcs.PNG
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KarishmaB
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  16 Feb 2012, 04:26
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enigma123 wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

Any idea how to solve this guys?


Given the ratio of minor arcs, their central angles and inscribed angles are in the same ratio.
arc BC: arc AB : arc AC = 1:2:6
So inscribed angles angle A : angle C : angle B = 1:2:6
Since these three angles form a triangle, their sum is 180 which means angle C = 40 degrees
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enigma123
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  11 Feb 2012, 15:26
How did you get x+2x+6x=360 Bunuel?
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  11 Feb 2012, 15:27
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enigma123 wrote:
How did you get x+2x+6x=360 Bunuel?


Circumference 360 degrees.
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  15 Feb 2012, 13:08
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Let BC \(=x\)
So AB \(=2x\)
So AC \(=6x\)

Now angle in question is opposite AB... and we know that Angles are in proportion to the length of the sides they are in front of..

We need to find \(2x\).....so ratio of this side(AB) \(=\frac{{2x}}{{9x}}=\frac{2}{9}\)

Sum of angles of triangle \(=180\)

So we can use ratios to setup an equation as follows:

\(\frac{{2x}}{{9x}}*180=40\)

Hence Angle is \(40\)
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  16 Feb 2012, 05:01
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lenght of the arc = (x/360)*2pir
Given AB=2BC; AC=3AB. If x is the angle of arc BC then AB=2x; AC=6x;
AB+BC+AC=360 --> x+2x+6x=360 --> 9x = 360 --> x = 40
Is this approach right? can someone help me please?
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  16 Feb 2012, 05:41
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pavan1mpv wrote:
lenght of the arc = (x/360)*2pir
Given AB=2BC; AC=3AB. If x is the angle of arc BC then AB=2x; AC=6x;
AB+BC+AC=360 --> x+2x+6x=360 --> 9x = 360 --> x = 40
Is this approach right? can someone help me please?


When you say, "If x is the angle of arc BC", you mean the central angle, right?
The sum of the central angles will be 360 so you get x = 40. But remember, this is the central angle subtended by arc BC. The central angle subtended by arc AB = 2x = 80.

But since we are interested in the inscribed angle (i.e. the angle on the circumference of the circle), we get angle BCA (inscribed angle of arc AB) = 80/2 = 40 degrees (since inscribed angle is half of central angle)
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  22 Feb 2012, 08:15
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as all arcs are given so we can add all arcs to find radius of the circle bcoz sum of all arcs is equal to circumference of the circle,

assume BC = X,
AB= 2X AND AC= 6X
implies 9X = 2pir
implies, r= 9X/2pi
now angle subtended by arc AB at the centre of circle is twice of the angle BCA.

arc AB = 2X = angle at centre of circle ( Q)/360 *2pir

implies 2X= Q/360*
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  22 Feb 2012, 08:18
sorry submitted uncompleted ,

2X= Q/360*2pi*9X/2PI
Gives us angle Q= 80
and angle BCA = 40 ( HALF OF THE ANGLE subtended at centre of circle)

answer B = 40 degrees
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  03 Jul 2015, 01:36
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Answer: B.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.


Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  03 Jul 2015, 02:17
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davesinger786 wrote:
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Answer: B.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.


Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.


Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  03 Jul 2015, 02:42
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Bunuel wrote:
davesinger786 wrote:
Bunuel wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Answer: B.

Check Circles chapter of Math Book for more: math-circles-87957.html

Hope it helps.


Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.


Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414


Thanks for that,Bunuel. So if I go by the logic that the diagram isn't accurate, arc AC subtends 240 at the center right? but in this question ,it's mentioned as the "minor arc AC" w.r.t to the calculations but going by the diagram in your article it looks like 240 is actually made by the major arc? Please help me get an idea of what are the angles subtended by the arcs ..getting confused here.Thanks in advance
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  03 Jul 2015, 11:39
davesinger786 wrote:
Bunuel wrote:
davesinger786 wrote:

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.


Check here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235232 and here: the-length-of-arc-axb-is-twice-the-length-of-arc-bzc-and-105748.html#p1235414


Thanks for that,Bunuel. So if I go by the logic that the diagram isn't accurate, arc AC subtends 240 at the center right? but in this question ,it's mentioned as the "minor arc AC" w.r.t to the calculations but going by the diagram in your article it looks like 240 is actually made by the major arc? Please help me get an idea of what are the angles subtended by the arcs ..getting confused here.Thanks in advance




That's true. If AC is larger than the two other archs, AC cannot be described as a minor arch. The archs are 2x their inscribed angles, and therefore, divide the degree of the arch by 2 to arrive at your answer.

Thanks,
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  14 Dec 2016, 15:59
enigma123 wrote:
The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

Any idea how to solve this guys?


what a nice question!
i got to the answer the same way
9x=360
x=40
2x=80
and as explained, the answer is B.
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  26 Aug 2017, 10:46
[quote="davesinger786"][quote="Bunuel"]The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA?
A. 20
B. 40
C. 60
D. 80
E. 120
Attachment:
Arcs.PNG

(arc AB) = 2*(arc BC) and (arc AC) = 3*(arc AB);

(arc BC)=x, (Arc AB)=2x, and (Arc AC)=3*2x=6x --> x+2x+6x=360 --> 9x=360 --> x=40 --> (arc AB)=2x=80 --> according to the central angle theorem the measure of inscribed angle (<BCA) is always half the measure of the central angle (<BOA, which is the minor arc AB), hence <BCA=80/2=40.

Answer: B.

Hi Bunuel,

I have a doubt here. If x =40, angle subtended by Arc Ac becomes 240.But how can arc AC subtend 240 at the center.If we join the center to A and C thereby making it a triangle, one angle itself exceeds 180 degrees.Please guide me where am I going wrong.



would be thankful if someone could explain the above quoted discrepancy in the problem
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  11 Aug 2018, 08:36
I think this question in incorrect because of the following case :-

Case 1: - when triangle inscribes centre of the circle within it.

In that case, x + 2x + 6x = 360

and 6x = 240. The arc subtended by angle 6x cannot be a minor arc, which is a main assumption in the question. Hence, this approach is incorrect.

Case 2:- when triangle does not inscribe circle within.

In that case, x + 2x = 6x, which is not possible.

There isn't any third case for this question. Therefore, I believe this question is incorrect.
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Re: The length of minor arc AB is twice the length of minor arc [#permalink] New post  30 Jul 2020, 10:31
Hi guys, I got the method to solve this question, but somehow I am messing up in the ratio and proportions part. Can someone correct me as to what exactly I am doing wrong:
Arc(AB)=2(Arc(BC) - equation 1 & Arc(AC)=3(Arc(AB)) - equation 2
I multiplied equation 1 by 3: 3Arc(AB)=6(Arc(BC) - equation 3
So the ratio I got by combining equations 2&3 was: Arc(AC)=3(Arc(AB))=6(Arc(BC), Which is AC:AB:BC=1:3:6
Cant seem to figure out where I am messing up. Help would be much appreciated. Thanks!!
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