A,B and C lie on a circle with center Q. The area of sector AQB is twice the area of sector BQC. The length of arc AB is 28 centimeters. What is the length, in centimeters, of arc BC ?

Question

A,B and  C lie on a circle with center  Q.  The area of sector  AQB is twice the area of sector  BQC. The length of arc  AB is 28 centimeters. What is the length, in centimeters, of  arc BC ?

Bytelearn · Accepted Answer

Identify Angle Relationship: Since the area of sector AQB is twice the area of sector BQC, the angle for sector AQB must be twice the angle for sector BQC. Assign Angle Measures: Let's call the angle for sector BQC x degrees. Then the angle for sector AQB is 2x degrees. Calculate Total Angle: The total angle for a circle is 360 degrees. So, the sum of the angles for sectors AQB and BQC is 360 degrees. Set Up Equation: We can write an equation: 2x + x = 360. Solve for Angle: Solving for x gives us x = 120 degrees. So, the angle for sector BQC is 120 degrees, and the angle for sector AQB is 240 degrees. Determine Arc Length: The length of arc AB is 28 centimeters, which corresponds to the angle of 240 degrees. Find Length of Arc BC: To find the length of arc BC, we need to find what length corresponds to 120 degrees. Set Up Proportion: We can set up a proportion since the lengths of arcs are proportional to their angles. Cross-Multiply: The proportion is 28 cm / 240 degrees = length of arc BC / 120 degrees. Solve for Arc BC: Cross-multiplying gives us (28 cm * 120 degrees) = (length of arc BC * 240 degrees). Calculate Arc BC Length: Solving for the length of arc BC gives us length of arc BC = (28 cm * 120 degrees) / 240 degrees. Calculating that gives us length of arc BC = (28 * 120) / 240 = 3360 / 240 = 14 centimeters.

Points $A, B$ and $C$ lie on a circle with center $Q$ . $\newline$ - The area of sector $\mathrm{AQB}$ is twice the area of sector $\mathrm{BQC}$ . $\newline$ - The length of arc $\mathrm{AB}$ is $28$ centimeters. $\newline$ What is the length, in centimeters, of $\operatorname{arc} B C$ ?

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