Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1959 AHSME Problems/Problem 32"

(diagram)
(→‎Solution)
 
Line 46: Line 46:
 
</asy>
 
</asy>
  
−
<math>\fbox{C}</math>
+
Let the circle have center <math>O</math>, let the point of tangency be point <math>T</math>, and let <math>B</math> be the intersection of <math>\overline{OA}</math> with the circle, as in the diagram. By the definitions of a circle and a tangent to a circle, we know that <math>OB=OT=r</math> and <math>\overline{OT} \perp \overline{TA}</math>. By the [[Pythagorean Theorem]], <math>OA=\sqrt{\frac{25r^2}{9}}=\frac{5}{3}r</math>. 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 <math>AB</math>. Because <math>OA=\frac{5}{3}r</math> and <math>OB=r</math>, <math>AB=\frac{5}{3}r-r=\frac{2}{3}r=\frac{(4/3)r}{2}=\boxed{\textbf{(C) }\frac{1}{2}l}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 21:15, 20 July 2024

Problem

The length $l$ of a tangent, drawn from a point $A$ to a circle, is $\frac43$ of the radius $r$. The (shortest) distance from A to the circle is: $\textbf{(A)}\ \frac{1}{2}r \qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \frac{1}{2}l\qquad\textbf{(D)}\ \frac23l \qquad\textbf{(E)}\ \text{a value between r and l.}$

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]

Let the circle have center $O$, let the point of tangency be point $T$, and let $B$ be the intersection of $\overline{OA}$ with the circle, as in the diagram. By the definitions of a circle and a tangent to a circle, we know that $OB=OT=r$ and $\overline{OT} \perp \overline{TA}$. By the Pythagorean Theorem, $OA=\sqrt{\frac{25r^2}{9}}=\frac{5}{3}r$. 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 $AB$. Because $OA=\frac{5}{3}r$ and $OB=r$, $AB=\frac{5}{3}r-r=\frac{2}{3}r=\frac{(4/3)r}{2}=\boxed{\textbf{(C) }\frac{1}{2}l}$.

See also

1959 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 31 		Followed by
Problem 33
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1959_AHSME_Problems/Problem_32&oldid=223239"