Video: SAT Practice Test 1 β€’ Section 4 β€’ Question 24

Circle 𝑂 is divided into 3 sectors. Points 𝐴, 𝐡, and 𝐢 are on the circumference of the circle. Sector 𝐴𝑂𝐢 has an area of 12πœ‹, and sector 𝐢𝑂𝐡 has an area of 18πœ‹. If the radius of the circle is 6, what is the measure of the central angle 𝐡𝑂𝐴 in degrees?

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Video Transcript

Circle 𝑂 is divided into three sectors. Points 𝐴, 𝐡, and 𝐢 are on the circumference of the circle. Sector 𝐴𝑂𝐢 has an area of 12πœ‹, and sector 𝐢𝑂𝐡 has an area of 18πœ‹. If the radius of the circle is six, what is the measure of the central angle 𝐡𝑂𝐴 in degrees?

It is sensible to start this question by sketching the circle 𝑂. Points 𝐴, 𝐡, and 𝐢 lie on the circumference such that the circle is split into three sectors. We’re told that the area of sector 𝐴𝑂𝐢 is 12πœ‹. And we’re also told that the area of sector 𝐢𝑂𝐡 is 18πœ‹. The radius of the circle is equal to six. Therefore, the lengths 𝑂𝐴, 𝑂𝐡, and 𝑂𝐢 are all equal to six. We’ve been asked to calculate the central angle 𝐡𝑂𝐴 in degrees, labelled πœƒ on the diagram.

The area of any circle can be calculated using the formula πœ‹π‘Ÿ squared, where π‘Ÿ is the radius of the circle. In our question, we need to multiply πœ‹ by six squared. Six squared is equal to 36. Therefore, we need to multiply πœ‹ by 36. This can be written as 36πœ‹.

As we already know the areas of sectors 𝐴𝑂𝐢 and 𝐢𝑂𝐡, we can calculate the area of sector 𝐡𝑂𝐴 by subtracting these values from 36πœ‹. The area of sector 𝐡𝑂𝐴 is equal to 36πœ‹ minus 12πœ‹ plus 18πœ‹. 12πœ‹ plus 18πœ‹ is equal to 30πœ‹. Therefore, we need to subtract 30πœ‹ from 36πœ‹. This gives us six πœ‹. The area of sector 𝐡𝑂𝐴 equals six πœ‹.

The area of any sector can be calculated using the formula πœƒ divided by 360 multiplied by πœ‹π‘Ÿ squared. πœ‹π‘Ÿ squared is the area of the whole circle, and πœƒ is the angle of the sector. We’ve already worked out that the area of the sector 𝐡𝑂𝐴 is equal to six πœ‹. The area of the whole circle, πœ‹π‘Ÿ squared, is equal to 36πœ‹. This means that six πœ‹ is equal to πœƒ divided by 360 multiplied by 36πœ‹.

We can divide both sides of this equation by πœ‹. This leaves us with six is equal to πœƒ divided by 360 multiplied by 36. The numerator and denominator of the right-hand side of this equation can be divided by 36. 36 divided by 36 is equal to one, and 360 divided by 36 is equal to 10. This means that six is equal to πœƒ divided by 10. Six multiplied by 10 is equal to 60. Therefore, the angle πœƒ equals 60 degrees.

We can therefore conclude that if sector 𝐴𝑂𝐢 has an area of 12πœ‹, sector 𝐢𝑂𝐡 has an area of 18πœ‹, and the radius of the circle is six, then the central angle 𝐡𝑂𝐴 will be equal to 60 degrees. We could also use this method to calculate the central angle of sectors 𝐴𝑂𝐢 and 𝐢𝑂𝐡.

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