Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

검증된 단계별 안내

Recall the formula for the length of an arc intercepted by a central angle: $L = r \theta$, where $L$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians.

Note that the central angle $\theta$ is fixed, so it remains constant in this problem.

If the radius $r$ is doubled, then the new radius becomes \$2r$.

Substitute the new radius into the arc length formula to find the new arc length: $L_{new} = (2r) \theta$.

Compare the new arc length $L_{new}$ to the original arc length $L$ to determine how the arc length changes when the radius is doubled.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Arc Length Formula

The arc length of a circle is calculated using the formula L = rθ, where r is the radius and θ is the central angle in radians. This formula shows that the arc length is directly proportional to both the radius and the angle.

Effect of Radius on Arc Length

Since arc length depends linearly on the radius, doubling the radius while keeping the central angle fixed will double the arc length. This means the arc length changes proportionally with the radius.

Central Angle in Radians

The central angle must be measured in radians for the arc length formula L = rθ to be valid. Radians relate the angle directly to the radius, making calculations of arc length straightforward and consistent.