1959 AHSME Problems/Problem 32
Problem
The length 
of a tangent, drawn from a point 
to a circle, is 
of the radius 
. The (shortest) distance from A to the circle is:
Solution
![[asy] import geometry; point O=(0,0); point A=(5,0); point B,T; circle c=circle(O,3); markscalefactor=0.05; // Circle, segment OA draw(c); dot(O); label("O",O,NW); dot(A); label("A",A,NE); draw(O--A); // Segments OT, OA line[] t1=tangents(c,A); pair[] t=intersectionpoints(t1[0], c); T=t[0]; dot(t[0]); label("T",T,SE); draw(A--T--O); draw(rightanglemark(A,T,O)); // Point B pair[] b=intersectionpoints((O--A),c); B=b[0]; dot("B",B,NE); // Length labels label("$r$",midpoint(O--T),SW); label("$r$",midpoint(O--B),N); label("$\frac{4}{3}r$",midpoint(A--T),SE); [/asy]](http://latex.artofproblemsolving.com/b/0/a/b0a7d663479391704b13abf3c42068b309b73d7f.png)
Let the circle have center 
, let the point of tangency be point 
, and let 
be the intersection of 
with the circle, as in the diagram. By the definitions of a circle and a tangent to a circle, we know that 
and 
. By the Pythagorean Theorem, 
. Because the shortest segment from an external point to a circle lies on the line connecting that point to the center of the circle, our desired distance is 
. Because 
and 
, 
.
See also
| 1959 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 31 |
Followed by Problem 33 | |
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