Question

Math

Posted 3 months ago

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In the figure below, $\overline{A C}$ and $\overline{B D}$ are diameters of circle $P$.
What is the arc measure of major arc B \overparen{D} C in degrees?
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Combine like terms in the equation $4k + 159 + 2k + 153 = 360$

step 2

Simplify to get $6k + 312 = 360$

step 3

Subtract 312 from both sides to isolate the term with $k$, resulting in $6k = 48$

step 4

Divide both sides by 6 to solve for $k$, yielding $k = 8$

step 5

Substitute $k = 8$ into the angle $APC$ to find its measure: angle $APC = 4k + 159$

step 6

Calculate the measure of angle $APC$ by substituting $k = 8$: angle $APC = 4(8) + 159 = 32 + 159 = 191$ degrees

step 7

Since $AC$ is a diameter, angle $APC$ intercepts a semicircle, and thus the arc measure of B\overparen{D}C is $360 - 191 = 169$ degrees

[question 1] Answer

The arc measure of major arc B \overparen{D} C is $169$ degrees.

Key Concept

Intercepted Arc by a Central Angle

Explanation

The measure of an arc intercepted by a central angle is equal to the measure of the angle itself. Since angle $APC$ intercepts a semicircle, the remaining arc B\overparen{D}C is the difference between $360$ degrees and the measure of angle $APC$.

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