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

Question

https://gmatclub.com/forum/download/file.php?id=16540 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 Arcs.PNG

Bunuel · Accepted Answer

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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Attachment:

Arcs.PNG [ 7.21 KiB | Viewed 40057 times ]

KarishmaB · Answer

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

enigma123 · Answer

How did you get x+2x+6x=360 Bunuel?

Bunuel · Answer

Circumference 360 degrees.

omerrauf · Answer

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

pavan1mpv · Answer

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?

KarishmaB · Answer

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)

pbull78 · Answer

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*

pbull78 · Answer

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

davesinger786 · Answer

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.

Bunuel · Answer

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

davesinger786 · Answer

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

mejia401 · Answer

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,

mvictor · Answer

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.

mrmocker · Answer

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.

Wonderwoman31 · Answer

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.

gauranggarg · Answer

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

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