0
$\begingroup$

Problem statement: Circle's point M is away from tangents A and B by 4 and 9 cm. Find how far away M is from a line AB.

My attempt: First I have drawn out the problem. Denoted x as the value that I need to find: Circle

I know only a few facts about inscribed triangles and one of them is the formula $S=\frac{abc}{4r}$. However it does not seem to be useful in this case as I don't really need a, b, c, r, nor S. I don't have the needed information to find them as well.

One thought that occurred to me was that I might accomplish something by searching for angles and then using sines' theorem. Problem - I don't know where to start.

Another idea that I got was to look at the fact that the problem points out how A and B are tangent lines. So, they make up right angle with the radius. Not sure where to go from there...

  • 3
    Your figure does not match the wording! The distance from $M$ to the tangent is by definition the distance from $M$ to the orthogonal projection of $M$ onto that tangent. – 2017-01-21
  • 0
    Your title question, which is utterly different from you body question, is unanswerable. As is the question in the body. Simply draw different circle. On a very large circle M will be close to AB. – 2017-01-21

1 Answers 1

1

In the corrected figure (see @Mercy King above), with MC = 4 perpendicular to the tangent at A, and MD= 9 perpendicular to the tangent at B, join MA and MB, and draw ME perpendicular to AB. Since triangles ACM and MEB are similar (see Euclid, Elements III, 32), $4/AM = ME/MB$. And since triangles AME and BMD are likewise similar, $AM/ME = MB/9$. Therefore, $4/ME = ME/9$, making $ME^2 = 36$ and $ME = 6$.

  • 0
    Where is C? Could you perhaps provide a drawing? – 2017-01-22
  • 0
    @Jens- C is on the tangent through A. MC = 4 is perpendicular to AC. D is on the tangent through B. MD = 9 is perpendicular to BD. Still learning how to draw my own figures. – 2017-01-22
  • 0
    You've convinced me. Well done! – 2017-01-22
  • 0
    With your permission, I can add a picture to your answer. – 2017-01-23
  • 0
    @Mick- Thank you, please do. – 2017-01-23