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4.7 G) Arc Length – Part 2
4.7 G) Arc Length – Part 2
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Example 1
The radius of the sector below is 5 cm. The exact length of the arc in the sector below is 8π. What is the angle for the sector?
The radius of the sector below is 5 cm. The exact length of the arc in the sector below is 8π. What is the angle for the sector?
We are going to answer this question by using the generic formula. We will then work backwards to find the angle of the sector. Here is the generic formula.
Let’s sub the values that we know into this equation. We are told that the arc length is 8π. We are also given the radius, which means that we are able to work out the diameter by doubling the radius; this means that the diameter is 10 (5 x 2). Let’s sub these values into the equation below and work through to find the angle.
There is a π on both sides of the equation. We are able to get rid of the π by dividing both sides of the equation by π.
The next step is to get rid of the “x 10” at the end of the equation. We are able to do this by doing the opposite; we divide both sides of the equation by 10.
At the moment the angle is being divided by 360. In order to get rid of the divide by 360, we multiply both sides of the equation by 360.
Therefore, the angle of the sector is 288°.
Example 2
A chord splits a circle into 2. The radius of the circle is 5 cm. The ratio of the minor arc length to the major arc is 1 : 4. What is the exact length of the major and minor arc?
A chord splits a circle into 2. The radius of the circle is 5 cm. The ratio of the minor arc length to the major arc is 1 : 4. What is the exact length of the major and minor arc?
This question does not require us to use the generic arc length formula. Instead, we answer it by finding the circumference of the whole circle and then we split it using the ratio 1 : 4. Let’s find the circumference of the whole circle and we do this by using the formula below:
We are asked to find the exact length, which means that we will give our answer in terms of π. The question tells us what the radius of the circle is, but the formula above uses the diameter. We know that the diameter of a circle is double the radius of a circle. This means that the diameter of the circle is 10 cm.
The circumference of the whole circle is 10π cm. We now need to split this up into the ratio 1 : 4. We do this by finding the total number of parts in the ratio and then find the amount that one part represents. There are 5 parts in our ratio (1 + 4). We have the following equation.
We want to know what one part of the ratio is, and we obtain what one part is by dividing by 5.
Therefore, one part in our ratio is equal to 2π. The final step is to multiply the number of parts that the minor and major arc have to see what their length is.
The minor arc has 1 part, which means that the length of the minor arc is 2π cm.
The major arc has 4 parts, which means that we need to multiply what one part is by 4, which tells us that the length of the major arc is 8π cm.
The minor arc has 1 part, which means that the length of the minor arc is 2π cm.
The major arc has 4 parts, which means that we need to multiply what one part is by 4, which tells us that the length of the major arc is 8π cm.