Question

The measure of Arc B A C is 240. What is the ratio of the measure of the major arc to the measure of the minor arc? 3:1 1:3 2:1 1:2.

Answer 1

The ratio of the measure of the major arc to the measure of the minor arc is given by: Option A: 3:1

What is ratio of two quantities?

Suppose that we've got two quantities with measurements as 'a' and 'b'

Then, their ratio(ratio of a to b) a:b or $\dfrac{a}{b}$

We usually cancel out the common factors from both the numerator and the denominator of the fraction we obtained. Numerator is the upper quantity in the fraction and denominator is the lower quantity in the fraction).

Suppose that we've got a = 6, and b= 4, then:

$a:b = 6:2 = \dfrac{6}{2} = \dfrac{2 \times 3}{2 \times 1} = \dfrac{3}{1} = 3\\or\\a : b = 3 : 1 = 3/1 = 3$

Remember that the ratio should always be taken of quantities with same unit of measurement. Also, ratio is a unit-less(no units) quantity.

How are arcs of a circle measured?

They are either measured by their length, or the angle they subtend on the center of the circle whom they belong to.

Remember that full circle(the circumference) is subtending complete angle, whose measure is $360^\circ$

What are major and minor arc and how are they related?

When two points are considered on the circumference of a specific circle, then the minor(smaller) part of the circumference is called minor arc, and the rest of the part of the circumference is called major arc. In case of them being equal, they are not minor or major but equal lengthed arcs.

If the arcs are measured in terms of angle they subtend on the center, then:

Angle subtended by minor arc + angle subtended by major arc = $360^\circ$

It is because their angles joined make complete angle, measuring $360^\circ$.

For the given case, referring to the figure attached below, we get:

Measure of arc BAC (in terms of angle they subtend) = 240 degrees.

Since BAC is major arc, thus, we get:

Measure of minor arc BC = $360^\circ - \text{Measure of major arc} = 360 - 240 = 120^\circ$

Their ratio is found as:

$\dfrac{M_{BAC}}{M_{BC}} = \dfrac{240}{120} = \dfrac{3}{1} = 3:1$

where M shows measurement of the arc written in subscript.

Thus, the ratio of the measure of the major arc to the measure of the minor arc is given by: Option A: 3:1

Learn more about arcs of a circle here:

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