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Difference between revisions of "2006 AMC 10A Problems/Problem 17"

m (+ imgs)
(+ <asy>s (v_enhance azjps))
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== Problem ==
 
== Problem ==
−
In rectangle <math>ADEH</math>, points <math>B</math> and <math>C</math> trisect <math>\overline{AD}</math>, and points <math>G</math> and <math>F</math> trisect <math>\overline{HE}</math>. In addition, <math>AH=AC=2</math>. What is the area of quadrilateral <math>WXYZ</math> shown in the figure?  
+
In [[rectangle]] <math>ADEH</math>, points <math>B</math> and <math>C</math> [[trisect]] <math>\overline{AD}</math>, and points <math>G</math> and <math>F</math> trisect <math>\overline{HE}</math>. In addition, <math>AH=AC=2</math>. What is the area of [[quadrilateral]] <math>WXYZ</math> shown in the figure?  
  
 
<math>\mathrm{(A) \ } \frac{1}{2}\qquad\mathrm{(B) \ } \frac{\sqrt{2}}{2}\qquad\mathrm{(C) \ } \frac{\sqrt{3}}{2}\qquad\mathrm{(D) \ } \frac{2\sqrt{2}}{2}\qquad\mathrm{(E) \ } \frac{2\sqrt{3}}{3}\qquad</math>
 
<math>\mathrm{(A) \ } \frac{1}{2}\qquad\mathrm{(B) \ } \frac{\sqrt{2}}{2}\qquad\mathrm{(C) \ } \frac{\sqrt{3}}{2}\qquad\mathrm{(D) \ } \frac{2\sqrt{2}}{2}\qquad\mathrm{(E) \ } \frac{2\sqrt{3}}{3}\qquad</math>
  
−
[[Image:2006_AMC10A-17.png]]
+
<!-- [[Image:2006_AMC10A-17.png]] -->
−
 
+
<asy>
 +
size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);
 +
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
 +
A=(0,2); B=(1,2); C=(2,2); D=(3,2);
 +
H=(0,0); G=(1,0); F=(2,0); E=(3,0);
 +
D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW);
 +
D(A--F); D(B--E); D(D--G); D(C--H);
 +
Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);
 +
D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
 +
D(A--D--E--H--cycle);
 +
</asy>
 
__TOC__
 
__TOC__
 
== Solution ==
 
== Solution ==
 
=== Solution 1 ===
 
=== Solution 1 ===
 
It is not difficult to see by [[symmetry]] that <math>WXYZ</math> is a [[square]].
 
It is not difficult to see by [[symmetry]] that <math>WXYZ</math> is a [[square]].
 +
<!-- [[Image:2006_AMC10A-17a.png]] -->
 +
<asy>
 +
size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);
 +
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
 +
A=(0,2); B=(1,2); C=(2,2); D=(3,2);
 +
H=(0,0); G=(1,0); F=(2,0); E=(3,0);
 +
D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW);
 +
D(A--F); D(B--E); D(D--G); D(C--H);
 +
Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);
 +
D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
 +
MP("3",(A+B)/2,2*N); MP("2",(A+H)/2,plain.W); D(B--Z); MP("1",(B+Z)/2,plain.W); MP("\frac{\sqrt{2}}{2}",(W+Z)/2,plain.SE);
 +
D(A--D--E--H--cycle);
 +
</asy>
 +
Draw <math>\overline{BZ}</math>. Clearly <math>BZ = \frac 12AH = 1</math>. Then <math>\triangle BWZ</math> is [[isosceles]], and is a <math>45-45-90 \triangle</math>. Hence <math>WZ = \frac{1}{\sqrt{2}}</math>, and <math>[WXYZ] = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac 12\ \mathrm{(A)}</math>.
  
−
[[Image:2006_AMC10A-17a.png]]
+
There are many different similar ways to come to the same conclusion using different [[right triangle|45-45-90 triangles]].
−
 
 
−
Draw <math>\overline{BZ}</math>. Clearly <math>BZ = \frac 12AH = 1</math>. Then <math>\displaystyle \triangle BWZ</math> is [[isosceles]], and is a <math>45-45-90 \triangle</math>. Hence <math>WZ = \frac{1}{\sqrt{2}}</math>, and <math>[WXYZ] = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac 12\ \mathrm{(A)}</math>.
 
−
 
 
−
There are many different similar ways to come to the same conclusion using different 45-45-90 triangles.
 
  
 
=== Solution 2 ===
 
=== Solution 2 ===
−
[[Image:2006_AMC10A-17b.png]]
+
<!-- [[Image:2006_AMC10A-17b.png]] -->
 +
<asy>
 +
size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10);
 +
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
 +
A=(0,2); B=(1,2); C=(2,2); D=(3,2);
 +
H=(0,0); G=(1,0); F=(2,0); E=(3,0);
 +
D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW);
 +
D(A--F); D(B--E); D(D--G); D(C--H);
 +
Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);
 +
D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
 +
D(B--D((A+H)/2)--G);D(C--D((E+D)/2)--F);
 +
D(A--D--E--H--cycle);
 +
</asy>
  
 
Draw the lines as shown above, and count the squares. There are 12, so we have <math>\frac{2\cdot 3}{12} = \frac 12</math>.
 
Draw the lines as shown above, and count the squares. There are 12, so we have <math>\frac{2\cdot 3}{12} = \frac 12</math>.
Line 26: Line 58:
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
[[Category:Area Problems]]

Revision as of 14:55, 20 August 2011

Problem

In rectangle $ADEH$, points $B$ and $C$ trisect $\overline{AD}$, and points $G$ and $F$ trisect $\overline{HE}$. In addition, $AH=AC=2$. What is the area of quadrilateral $WXYZ$ shown in the figure?

$\mathrm{(A) \ } \frac{1}{2}\qquad\mathrm{(B) \ } \frac{\sqrt{2}}{2}\qquad\mathrm{(C) \ } \frac{\sqrt{3}}{2}\qquad\mathrm{(D) \ } \frac{2\sqrt{2}}{2}\qquad\mathrm{(E) \ } \frac{2\sqrt{3}}{3}\qquad$

[asy] size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2); H=(0,0); G=(1,0); F=(2,0); E=(3,0); D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW); D(A--F); D(B--E); D(D--G); D(C--H); Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W); D(A--D--E--H--cycle); [/asy]

Solution

Solution 1

It is not difficult to see by symmetry that $WXYZ$ is a square. [asy] size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2); H=(0,0); G=(1,0); F=(2,0); E=(3,0); D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW); D(A--F); D(B--E); D(D--G); D(C--H); Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W); MP("3",(A+B)/2,2*N); MP("2",(A+H)/2,plain.W); D(B--Z); MP("1",(B+Z)/2,plain.W); MP("\frac{\sqrt{2}}{2}",(W+Z)/2,plain.SE); D(A--D--E--H--cycle); [/asy] Draw $\overline{BZ}$. Clearly $BZ = \frac 12AH = 1$. Then $\triangle BWZ$ is isosceles, and is a $45-45-90 \triangle$. Hence $WZ = \frac{1}{\sqrt{2}}$, and $[WXYZ] = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac 12\ \mathrm{(A)}$.

There are many different similar ways to come to the same conclusion using different 45-45-90 triangles.

Solution 2

[asy] size(7cm); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2); H=(0,0); G=(1,0); F=(2,0); E=(3,0); D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,SE); D('F',F,SE); D('G',G,SW); D('H',H,SW); D(A--F); D(B--E); D(D--G); D(C--H); Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W); D(B--D((A+H)/2)--G);D(C--D((E+D)/2)--F); D(A--D--E--H--cycle); [/asy]

Draw the lines as shown above, and count the squares. There are 12, so we have $\frac{2\cdot 3}{12} = \frac 12$.

See Also

2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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
All AMC 10 Problems and Solutions
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