Lengths of arcs AP, BP are 20, 16 respectively, as shown. Find measure of angle AXP.

[ssolaming]

Can anyone solve this for me? The answer is 10 degree, but I don't know why

 

[Dr.Peterson]

First, determine the central angles for the two arcs, using the fact that AB is a diameter.

Then see this page for the appropriate theorem: https://mathbitsnotebook.com/Geometry/Circles/CRAngles.html

 

[pka]

View attachment 11527
Can anyone solve this for me? The answer is 10 degree, but I don't know why

Because the line $\displaystyle \overleftrightarrow {XP}$ is a tangent to the circle $\displaystyle O$ so the the angle tex]m(\algleOPX)=\frac{\pi}{2}[/tex] or a right angle. Now consider $\displaystyle angle POB$ intercepts an arc of length $\displaystyle 16$, Can ypu use $\displaystyle \frac{36}{180^o}=\frac{16}{m(\angle POB)}$ to show that $\displaystyle m(\angle POB)=80^o$ so $\displaystyle m(\angle PXB)=10^o$.