Ratio between concentric arcs | Trigonometry | Khan Academy

TL;DR

The video explains how to determine the ratio of arc lengths in a circle based on the lengths of their corresponding segments.

Transcript

In this diagram we have two arcs centered at point p. Arc AB. And you could imagine this arc could be part of a circle like this that is centered at point AB. And then you have this larger arc cd that would be part of a larger circle, just like that, centered at point P. And we don't know the actual angle measure of angle P. Right over here. But wi... Read More

Key Insights

• ✖️ Arc lengths can be determined by multiplying the angle measure (in radians) by the radius of a circle.
• 🥳 The ratio of arc lengths is independent of the actual angle measure but can be calculated using the ratio of their corresponding radii.
• 🥳 The ratio of arc lengths remains constant as long as the ratio of radius lengths is consistent.

Questions & Answers

Q: How can the length of an arc be calculated without knowing the angle measure?

In a circle, the length of an arc can be found by multiplying the angle measure (in radians) by the radius. This is because angle measure in radians reflects the number of radii intercepted by the arc.

Q: What is the ratio of the lengths of arc AB and arc CD in this diagram?

The ratio of the lengths of arc AB and arc CD is 5:9. This can be derived by considering the radius lengths, which are 5 units for arc AB and 9 units for arc CD.

Q: If the length of arc AB is given as 25/8, what other information can be determined?

With the knowledge that the length of arc AB is 25/8, we can ascertain the angle measure in radians and the length of arc CD. By solving for x in the equation 5x = 25/8, we find x = 5/8, indicating the angle measure of 5/8 radians. To find the length of arc CD, we multiply 9 (length of each radius) by 5/8, resulting in 45/8.

Q: How is the ratio of arc lengths affected by changes in the radius length?

The ratio of arc lengths remains constant regardless of the radius length. As long as the same ratio of radius lengths is maintained, the ratio of arc lengths will be unchanged.

Summary & Key Takeaways

• The video discusses two arcs in a circle, one smaller (AB) and one larger (CD), and aims to find the ratio of their lengths.

• Without knowing the angle measure, the video explains that the length of an arc can be calculated by multiplying the angle measure (in radians) by the radius.

• By applying this concept, the video determines that the ratio of the lengths of arc AB to arc CD is 5:9.