Difference between revisions of "2005 AMC 12A Problems/Problem 15"

Revision as of 21:37, 13 June 2018

Problem

Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$ . Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$ ?

$[asy] unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$

$(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}$

Solution

Solution 1

Notice that the bases of both triangles are diameters of the circle. Hence the ratio of the areas is just the ratio of the heights of the triangles, or $\frac{CD}{CF}$ ( $F$ is the foot of the perpendicular from $C$ to $DE$ ).

Call the radius $r$ . Then $AC = \frac 13(2r) = \frac 23r$ , $CO = \frac 13r$ . Using the Pythagorean Theorem in $\triangle OCD$ , we get $\frac{1}{3}r^2 + CD^2 = r^2 \Longrightarrow CD = \frac{2\sqrt{2}}3r$ .

Now we have to find $CF$ . Notice $\triangle OCD \sim \triangle OFC$ , so we can write the proportion:

$\frac{OF}{OC} = \frac{OC}{OD}$
$\frac{OF}{\frac{1}{3}r} = \frac{\frac{1}{3}r}{r}$
$OF = \frac 19r$

By the Pythagorean Theorem in $\triangle OFC$ , we have $\left(\frac{1}{9}r\right)^2 + CF^2 = \left(\frac{1}{3}r\right)^2 \Longrightarrow CF = \sqrt{\frac{8}{81}r^2} = \frac{2\sqrt{2}}{9}r$ .

Our answer is $\frac{CD}{CF} = \frac{\frac{2\sqrt{2}}{3}r}{\frac{2\sqrt{2}}{9}r} = \frac 13 \Longrightarrow \mathrm{(C)}$ .

Solution 2

Let the center of the circle be $O$ .

Note that $2 \cdot AC = BC \Rightarrow 3 \cdot AC = AB$ .

$O$ is midpoint of $AB \Rightarrow \frac{3}{2}AC = AO \Rightarrow CO = \frac{1}{3}AO \Rightarrow CO = \frac{1}{6} AB$ .

$O$ is midpoint of $DE \Rightarrow$ Area of $\triangle DCE = 2 \cdot$ Area of $\triangle DCO = 2 \cdot (\frac{1}{6} \cdot$ Area of $\triangle ABD) = \frac{1}{3} \cdot$ Area of $\triangle ABD \Longrightarrow \mathrm{(C)}$ .

Solution 3

Let $r$ be the radius of the circle. Note that $AC+BC = 2r$ so $AC = \frac{2}{3}r$ .

By Power of a Point Theorem, $CD^2= AC \cdot BC = 2\cdot AC^2$ , and thus $CD = \sqrt{2} \cdot AC = \frac{2\sqrt{2}}{3}r$

Then the area of $\triangle ABD$ is $\frac{1}{2} AB \cdot CD = \frac{2\sqrt{2}}{3}r^2$ . Similarly, the area of $\triangle DCE$ is $\frac{1}{2}(r-AC) \cdot 2 \cdot CD = \frac{2\sqrt{2}}{9}r^2$ , so the desired ratio is $\frac{\frac{2\sqrt{2}}{9}r^2}{\frac{2\sqrt{2}}{3}r^2} = \frac{1}{3} \Longrightarrow \mathrm{(C)}$

Solution 4

unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
pair D=dir(aCos(C.x)), e=(-D.x,-D.y);
pair H=(e.x,0);
draw(A--B--D--cycle);
draw(D--e--C);
draw(unitcircle,white);
drawline(D,C);
dot(O);
clip(unitcircle);
draw(unitcircle);
label("$E$",e,SSE);
label("$B$",B,NE);
label("$A$",A,W);
label("$D$",D,NNW);
label("$C$",C,SW);
label("$H$",H,SE);
draw(e--H,dashed);
label("O",(0,0),NE);
label("1",(C--O),N);
label("1",(H--O),S);
label("2",(A--C),N);
label("3",(O--D),NE);
label("3",(O--e),NE);
label("$2\sqrt{2}$",(D--C),NW);
label("$2\sqrt{2}$",(H--e),E)
draw(rightanglemark(e,(e.x,0),A,2));
draw(rightanglemark(D,C,B,2)); (Error making remote request. Unknown error_msg)

Let the center of the circle be $O$ . Without loss of generality, let the radius of the circle be equal to $3$ . Thus, $AO=3$ and $OB=3$ . As a consequence of $2(AC)=BC$ , $AC=2$ and $CO=1$ . Also, we know that $DO$ and $OE$ are both equal to $3$ due to the fact that they are both radii. Thus from the Pythagorean Theorem, we have DC being equal to $\sqrt{3^2-1^2}$ or $2\sqrt{2}$ . Now we know that the area of $[ABD]$ is equal to $\frac{(3+2+1)(2\sqrt{2})}{2}$ or $6\sqrt{2}$ . Know we need to find the area of $[DCE]$ . By simple inspection $[COD]$ /cong $[HOE]$ due to angles being equal and CPCTC. Thus $HE=3$ and $OH=1$

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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