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

[furtadovinod]

Hi,

Not posting the diagram here cause I am not sure how to. The problem states that

The length of minor arc AB is twice the length of minor arc BC and the length of major arc AC is three times the length of minor arc AB. What is the measure of angle BCA?

It seems to me that the bolded portion must read "the length of minor arc ......"

Pls clarify.

[venkat_yj]

Don't think the question is wrong.
Without the benefit of diagram, here is my proposed solution-

Given arc lengths-
AB = 2 x BC
AC = 3 x AB

Assuming BC = y, AB= 2y, AC = 6y.

Circumference = AB+BC+AC = 9y
arc y has an angle of 360/9 = 40 degrees

Length of arc BCA = BC+AC = y+6y = 7y

Therefore angle BCA = 7 x 40 = 280 degrees.

[nehag84]

Pls refer to the link given below...

post27452.html?hilit=The%20length%20of%20minor%20arc#p27452

Regards,
NG

[venkat_yj]

Thanks for the link to the diagram. I misunderstood the question. I was assuming angle BCA was referring to the angle subtended by the arc BCA at the center of the circle. Didn't realize the diagram was referring to a triangle.

With all the calculations remaining the same as my previous post, angle BCA is equal to half the angle subtended by the opposite arc (AB) at the center of the circle.

AB = 2y = 80, so angle BCA = 40.

PS: Its imp to remember that the diagram is not to scale unless specified. In this case, the posted diagram shows that all arcs are almost of same length; and that might throw a few of us off track. Also arc AC has two paths on the circle (assume you are biking this trail) - one clockwise and the other counter-clockwise. So arc AC does not have to be equal to AB+BC when it comes to circles or any closed loop diagrams. These are kind of questions that lead to careless mistakes.

[furtadovinod]

Thanks for the link to the diagram. I misunderstood the question. I was assuming angle BCA was referring to the angle subtended by the arc BCA at the center of the circle. Didn't realize the diagram was referring to a triangle.

With all the calculations remaining the same as my previous post, angle BCA is equal to half the angle subtended by the opposite arc (AB) at the center of the circle.

AB = 2y = 80, so angle BCA = 40.

PS: Its imp to remember that the diagram is not to scale unless specified. In this case, the posted diagram shows that all arcs are almost of same length; and that might throw a few of us off track. Also arc AC has two paths on the circle (assume you are biking this trail) - one clockwise and the other counter-clockwise. So arc AC does not have to be equal to AB+BC when it comes to circles or any closed loop diagrams. These are kind of questions that lead to careless mistakes.

Hi Venkat,

Pls note that for PS, diagrams ARE drawn to scale unless specified.

[Ben Ku]

Thanks venkat_yj, for providing a good approach to this problem.

If we see that the ratio of arcs AC:AB:BC is 6:2:1, we can say arc AC = 6x, arc AB = 2x, and arc BC = x
arc AC + arc AB + arc BC = 360
6x + 2x + x = 360
9x = 360
x = 40

Because the measure of an inscribed angle is one half the measure of its intercepted arc,

angle BCA = 1/2 (arc AB)= 1/2 (2x) = 1/2 (80) = 40

In Quant problems, all diagrams are draw to scale, unless otherwise indicated. In this problem, the figure was noted: "Figure not drawn to scale."

[Kweku.Amoako]

AB = 2BC
AC = 3BC

OR AC = 3AB = 6BC

some imagination is reuired here. if you redraw the diagram such that you have a line from the center of the circle(call it point P) to each point (A,B and C) then

AB is equivalent to APB
BC is equivalent to BPC
AC is equivalent to APC

notice that on this same diagram, if you insert the orignal lines in the diagram which are lines AB, BC and AC you will realize that

BCA = APB / 2

so if we can find APB then we can find BCA

now we already know that AC = 3AB = 6BC which is equivalent to APC = 3APB = 6 BPC. We also know that from circle theory that APB + BPC +APC = 360 . Therefore using ratios if we make AB = x then AC = 3x and BC = 0.5x

x + 3x + 0.5 x = 360

4.5x = 360

x = 80
pugging this back into the equation BCA = APB / 2 since we found AB = APB = 80

then BCA = 80 /2 = 40

[Ben Ku]

Thanks kweku, that works too.