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Difference between revisions of "1997 AHSME Problems/Problem 19"

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From the diagram above, it is more direct to note that BC = CF + BF = r - <math>\sqrt{3}</math> + r - 1 = 2
 
From the diagram above, it is more direct to note that BC = CF + BF = r - <math>\sqrt{3}</math> + r - 1 = 2
 +
 +
== Solution 3 ==
 +
The total area of both kites is <math>BC \cdot OF= 2r</math>. Thus, <math>r^2 = \frac{\sqrt{3}}{2} + 2r</math>
  
 
== See also ==
 
== See also ==
 
{{AHSME box|year=1997|num-b=18|num-a=20}}
 
{{AHSME box|year=1997|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:28, 10 October 2024

Problem

A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?


[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW); label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]

$\textbf{(A)}\ 2.18\qquad\textbf{(B)}\ 2.24\qquad\textbf{(C)}\ 2.31\qquad\textbf{(D)}\ 2.37\qquad\textbf{(E)}\ 2.41$

Solution

[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); pair D = (2.33,0); pair E = (0, 2.33); pair F = (0.35, 1.1); dot(A);dot(B);dot(C);dot(O);dot(D);dot(E);dot(F); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW);label("$D$",D,SW);label("$E$",E,SW);label("$F$", F, W); label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C);draw(O--E);draw(O--D); draw(Arc(O,2.33,163,288.5));[/asy]

Draw radii $OE$ and $OD$ to the axes, and label the point of tangency to triangle $ABC$ point $F$. Let the radius of the circle $O$ be $r$. Square $OEAD$ has side length $r$.

Because $BD$ and $BF$ are tangents from a common point $B$, $BD = BF$.

$AD = AB + BD$

$r = 1 + BD$

$r = 1 + BF$

Similarly, $CF = CE$, and we can write:

$AE = AC + CE$

$r = \sqrt{3} + CF$

Equating the radii lengths, we have $1 + BF = \sqrt{3} + CF$

This means $BF - CF = \sqrt{3} - 1$

$BF + CF = 2$ by the 30-60-90 triangle.

Therefore, $2BF = 2 + \sqrt{3} - 1$, and we get $BF = \frac{1}{2} + \frac{\sqrt{3}}{2}$

The radius of the circle is $AD$, which is $BF + 1 = \frac{3}{2} + \frac{\sqrt{3}}{2}$

Using decimal approximations, $r \approx 1.5 + \frac{1.73^+}{2} \approx 2.37$, and the answer is $\boxed{D}$.

Solution 2


From the diagram above, it is more direct to note that BC = CF + BF = r - $\sqrt{3}$ + r - 1 = 2

Solution 3

The total area of both kites is $BC \cdot OF= 2r$. Thus, $r^2 = \frac{\sqrt{3}}{2} + 2r$

See also

1997 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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
All AHSME Problems and Solutions

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