A 60 degree arc, of circle A, has exactly the same lenght as a 45 degree arc, of circle B. Find the ratio of the radius of circle A to the radius of circle B
To find the ratio of the radius of circle A to the radius of circle B, we can start by understanding the relationship between the angle of an arc and its length.
The general formula to calculate the length of an arc is given by L = (θ/360) * 2πr, where L represents the arc length, θ is the central angle in degrees, π is a mathematical constant approximately equal to 3.14159, and r represents the radius of the circle.
Let's start by finding the length of the 60-degree arc in circle A. Using the formula, we have:
L_A = (60/360) * 2πr_A
= (1/6) * 2πr_A
= (π/3) * r_A
Similarly, for the 45-degree arc in circle B, we have:
L_B = (45/360) * 2πr_B
= (1/8) * 2πr_B
= (π/4) * r_B
Given that L_A = L_B, we can set the two arc length equations equal to each other and solve for the ratio of the radii:
(π/3) * r_A = (π/4) * r_B
We can cancel out the π and rearrange the equation to solve for the ratio:
r_A/r_B = (1/4)/(1/3)
= 3/4
Therefore, the ratio of the radius of circle A to the radius of circle B is 3/4.
Ar/Br = 45/60 = 3/4.
Ar = Radius of circle A.
Br = Radius of circle b.
Circle A is the smaller circle.