Final answer:
To find mMP⌢, add the expressions for arc MN⌢ and arc NP⌢ and set the sum equal to 180° because they are part of a semicircle. Solve for x and then subtract mNP⌢ from 180° to find mMP⌢. The result is mMP⌢ = 86.75°.
Step-by-step explanation:
To find the measure of arc MP⌢ (mMP⌢), we must first understand that the diameter of a circle, such as line NQ in ⊙R, creates a semicircle. Thus, the sum of the measures of arc MN⌢ and arc NP⌢, which are part of the semicircle, is 180 degrees. Given mMN⌢=(6x+23)° and mNP⌢=(−13+10x)°, we can set up the following equation:
6x + 23 + (-13 + 10x) = 180
Simplifying the equation gives us:
16x + 10 = 180
Subtracting 10 from both sides:
16x = 170
Dividing both sides by 16:
x = 170 / 16
x = 10.625
Now that we have the value of x, we can find the measure of each arc. Since we need mMP⌢:
mMP⌢ = 180 - mNP⌢
Substitute the expression for mNP⌢ (−13 + 10x) and the value of x:
mMP⌢ = 180 - (−13 + 10(10.625))
Calculate:
mMP⌢ = 180 - (−13 + 106.25)
mMP⌢ = 180 - 93.25
mMP⌢ = 86.75°
Therefore, mMP⌢ is 86.75 degrees.