Exploring the Ratio of Radii in Circles with Equal Arc Lengths

In this article, we will delve into the fascinating world of circle geometry and explore how to determine the ratio of radii of two circles given that the arcs of the same length subtend angles of different measures at their centers. This problem requires an understanding of arc length and the relationship between angles and radii in circles.

Understanding the Problem

Consider two circles with equal arc lengths subtending angles of 60° and 90° at their respective centers. Our objective is to find the ratio of their radii. We will achieve this using the formula for the length of an arc:

Arc Length Formula

The formula for the length of an arc L in a circle is given by:

[L r theta]

Where:

L is the length of the arc. r is the radius of the circle. (theta) is the angle subtended by the arc in radians.

Converting Degrees to Radians

First, we need to convert the given angles from degrees to radians. The conversion factor is:

[text{radians} text{degrees} times frac{pi}{180}]

For the 60° angle:

[theta_1 60^circ times frac{pi}{180} frac{pi}{3} text{ radians}]

For the 90° angle:

[theta_2 90^circ times frac{pi}{180} frac{pi}{2} text{ radians}]

Let r_1 be the radius of the circle with the 60° angle, and r_2 be the radius of the circle with the 90° angle. The lengths of the arcs can be expressed as:

[L r_1 cdot frac{pi}{3} quad text{for the first circle}]

[L r_2 cdot frac{pi}{2} quad text{for the second circle}]

Since the lengths of the arcs are equal, we set them equal to each other:

[r_1 cdot frac{pi}{3} r_2 cdot frac{pi}{2}]

We can cancel (pi) from both sides:

[r_1 cdot frac{1}{3} r_2 cdot frac{1}{2}]

Now, rearranging gives:

[frac{r_1}{r_2} frac{frac{1}{2}}{frac{1}{3}} frac{1}{2} times frac{3}{1} frac{3}{2}]

Final Answer

Thus, the ratio of the radii (r_1) to (r_2) is:

[frac{r_1}{r_2} frac{3}{2}]

Generalizing the Solution

Now, let's generalize the problem for any given angles. Suppose the arcs of equal lengths (L) subtend angles of 65° and 110° at the centers of two circles with radii (r_1) and (r_2), respectively. The arc length formula gives us:

[L r_1 times frac{65 pi}{180}] [L r_2 times frac{110 pi}{180}]

Equating the right-hand sides:

[r_1 times frac{65 pi}{180} r_2 times frac{110 pi}{180}]

Cancelling (pi) from both sides:

[r_1 times 65 r_2 times 110]

Rearranging for the ratio of the radii:

[frac{r_1}{r_2} frac{110}{65} frac{22}{13}]

Conclusion

In this article, we have explored how to find the ratio of radii in circles with equal arc lengths subtending different angles at their centers. By understanding the relationship between arc length, angles, and radii, we can solve a wide range of circle geometry problems. The key takeaways include:

Converting angles from degrees to radians using the conversion factor (frac{pi}{180}). Using the arc length formula to relate arc length to radius and angle. Solving for the ratio of radii through algebraic manipulation.

This knowledge is valuable in various fields, including mathematics, engineering, and physics, and can be applied to real-world scenarios involving circular objects and their properties.