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Say there is a circle P. The line XE is tangent to circle P, with E being the point of tangency. There is the line XP. The measures of angle EXP is 30º, and the measure of arc EW (W is the second point that the line XP intersects the circle reading from left to right) is 85º. Arc x is the arc EV (where V is the first point that the line XP intersects the circle reading from left to right). I tried three methods of finding x. enter image description here

For the first method, I used the fact that the measures of angles not on the circle are equal to difference of the arcs they intercept divided by 2. For the second method I used the fact that there is a semicircle. 85º + x = 180. Thus, x is 95º. For the third, I drew a radius and used the properties of tangents. Angle XPE must be 90º because radii and tangents are perpendicular to each other (for a given circle). Then, angle EXP and angle XPE must be complementary, making angle XPE 60º and thus x 60º (because the measures of central angles equals the measures of the corresponding arcs).

Where is my logic going wrong? Why do I have three totally different answers? Note: this is not a homework question. This is a problem I got on a test a while ago and randomly found again. When I tried to solve it, I couldn’t figure it out. Thank you!

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    $\begingroup$ How could arc $EW$ measure $85$ degrees? I assume you mean the measure of $\angle \,EPW$, yes? It's clearly greater than a quarter of the circumference, hence greater than $90$ degrees... Anyway, $\angle \, EPX=60$ by simple angle chasing, which makes $\angle EPW =120$. $\endgroup$ Commented yesterday
  • $\begingroup$ @lulu Oh wow—I would have never realized that that was what was wrong…thank you so so much!!! $\endgroup$ Commented yesterday

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  1. is okay if we ignore that $P$ goes through the center of the circle.

But if $P$ goes through the center we cant have $EPW=85$. We must have $EPW=180-(90-EXP)=90+EXW$. so $EXW=30$ and $EPW=85$ is impossible. In fact $EPW\le 90$ is impossible.

  1. is okay if we ignore everything about the point $X$ and the angle $EXW$.

But if we we have $EPW = 85$ we can't have $EXW=30$. In fact, the angle $EXW$ and the point $X$ can't actually exist with $x \ge 90$.

  1. is fine if we ignore the arc $EPW$.

But To include the arc, if $EXW = 30$ then $EPW$ would have to equal exactly $120$ and having it equal $85$ is impossible.

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THis is simply a case of having inconsistant data. Or in other words, angle $EXP$ and arc $EPW$ are dependent upon each other (if we assume the line intersects the center) so we can't assign them arbitrary and independent values.

(If we do have two arbitrary independent values for them, then where the line passes in the circle is dependent on those values and we can't assume the line passes through the center.)

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  • $\begingroup$ Thank you so much!! This breakdown is very helpful! $\endgroup$ Commented yesterday

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