Difference between revisions of "2008 AMC 10B Problems/Problem 24"

(Solution: + sol)
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==Solution==
 
==Solution==
 
+
===Solution 1 ===
 
Draw the angle bisectors of the angles <math>ABC</math> and <math>BCD</math>. These two bisectors obviously intersect. Let their intersection be <math>P</math>.  
 
Draw the angle bisectors of the angles <math>ABC</math> and <math>BCD</math>. These two bisectors obviously intersect. Let their intersection be <math>P</math>.  
 
We will now prove that <math>P</math> lies on the segment <math>AD</math>.  
 
We will now prove that <math>P</math> lies on the segment <math>AD</math>.  
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label("$85^\circ$",C + .5*dir(120+42.5));
 
label("$85^\circ$",C + .5*dir(120+42.5));
 
label("$85^\circ$",C + .5*dir(120+85+42.5));
 
label("$85^\circ$",C + .5*dir(120+85+42.5));
 +
</asy>
 +
 +
=== Solution 2 ===
 +
Draw the diagonals <math>\overline{BD}</math> and <math>\overline{AC}</math>, and suppose that they intersect at <math>E</math>. Then, <math>\triangle ABC</math> and <math>\triangle BCD</math> are both isosceles, so by angle-chasing, we find that <math>\angle BAC = 55^{\circ}</math>, <math>\angle </math>CBD = 5^{\circ}<math>, and </math>\angle BEA = 180 - \angle EBA - \angle BAE = 60^{\circ}<math>. Draw </math>E'<math> such that </math>EE'B = 60^{\circ}<math> and so that </math>E'<math> is on </math>\overline{AE}<math>, and draw </math>E''<math> such that </math>\angle EE''C = 60^{\circ}<math> and </math>E''<math> is on </math>\overline{DE}<math>. It follows that </math>\triangle BEE'<math> and </math>\triangle CEE''<math> are both equilateral. Also, it is easy to see that </math>\triangle BEC \cong \triangle DE''C<math> and </math>\triangle BCE \cong \triangle BAE'<math> by construction, so that </math>DE'' = BE = EE'<math> and </math>EE'' = CE = E'A<math>. Thus, </math>AE = AE' + E'E = EE'' + DE'' = DE<math>, so </math>\triangle ADE<math> is isosceles. Since </math>\angle AED = 120^{\circ}<math>, then </math>\angle DAC = \frac{180 - 120}{2} = 30^{\circ}<math>, and </math>\angle BAD = 30 + 55 = 85^{\circ}$.
 +
<asy>
 +
import graph; size(6.73cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.237,xmax=2.492,ymin=-0.16,ymax=1.947;
 +
pen evefev=rgb(0.898,0.937,0.898), qqwuqq=rgb(0,0.392,0);
 +
filldraw(arc((1,0),0.141,115,175)--(1,0)--cycle,evefev,qqwuqq); filldraw(arc((0.658,0.94),0.051,175,235)--(0.658,0.94)--cycle,evefev,qqwuqq); draw((0,0)--(1,0)); draw((1,0)--(0.658,0.94)); draw((0.658,0.94)--(0.158,1.806)); draw((0.158,1.806)--(0,0)); draw((0,0)--(0.658,0.94)); draw((0.158,1.806)--(1,0)); draw((0.058,0.082)--(1,0)); draw((0.558,0.948)--(0.658,0.94));
 +
dot((0,0),ds); label("$A$",(-0.096,0.005),NE*lsf); dot((1,0),ds); label("$B$",(1.117,0.028),NE*lsf); dot((0.658,0.94),ds); label("$C$",(0.727,0.996),NE*lsf); dot((0.158,1.806),ds); label("$D$",(0.187,1.914),NE*lsf); dot((0.6,0.857),ds); label("$E$",(0.479,0.825),NE*lsf); dot((0.058,0.082),ds); label("$E'$",(0.1,0.23),NE*lsf); label("$60^\circ$",(0.767,0.091),NE*lsf,qqwuqq); dot((0.558,0.948),ds); label("$E''$",(0.423,0.957),NE*lsf); label("$60^\circ$",(0.761,0.886),NE*lsf,qqwuqq);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 
</asy>
 
</asy>
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=23|num-a=25}}
 
{{AMC10 box|year=2008|ab=B|num-b=23|num-a=25}}

Revision as of 12:06, 26 August 2010

Problem

Quadrilateral $ABCD$ has $AB = BC = CD$, angle $ABC = 70$ and angle $BCD = 170$. What is the measure of angle $BAD$?

$\mathrm{(A)}\ 75\qquad\mathrm{(B)}\ 80\qquad\mathrm{(C)}\ 85\qquad\mathrm{(D)}\ 90\qquad\mathrm{(E)}\ 95$

Solution

Solution 1

Draw the angle bisectors of the angles $ABC$ and $BCD$. These two bisectors obviously intersect. Let their intersection be $P$. We will now prove that $P$ lies on the segment $AD$.

Note that the triangles $ABP$ and $CBP$ are equal, as they share the side $BP$, and we have $AB=BC$ and $\angle ABP = \angle CBP$.

Also note that for similar reasons the triangles $CBP$ and $CDP$ are equal.

Now we can compute their inner angles. $BP$ is the bisector of the angle $ABC$, hence $\angle ABP = \angle CBP = 35^\circ$, and thus also $\angle CDP = 35^\circ$. $CP$ is the bisector of the angle $BCD$, hence $\angle BCP = \angle DCP = 85^\circ$, and thus also $\angle BAP = 85^\circ$.

It follows that $\angle APB = \angle BPC = \angle CPD = 180^\circ - 35^\circ - 85^\circ = 60^\circ$. Thus the angle $APB$ has $180^\circ$, and hence $P$ does indeed lie on $AD$. Then obviously $\angle BAD = \angle BAP = \boxed{ 85^\circ }$.

[asy] unitsize(1cm); defaultpen(.8); real a=4; pair A=(0,0), B=a*dir(0), C=B+a*dir(110), D=C+a*dir(120); draw(A--B--C--D--cycle); pair P1=B+3*a*dir(145), P2=C+3*a*dir(205); pair P=intersectionpoint(B--P1,C--P2); draw(B--P--C); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,N); label("$P$",P,W);  label("$35^\circ$",B + dir(180-17.5)); label("$35^\circ$",B + dir(180-35-17.5));  label("$85^\circ$",C + .5*dir(120+42.5)); label("$85^\circ$",C + .5*dir(120+85+42.5)); [/asy]

Solution 2

Draw the diagonals $\overline{BD}$ and $\overline{AC}$, and suppose that they intersect at $E$. Then, $\triangle ABC$ and $\triangle BCD$ are both isosceles, so by angle-chasing, we find that $\angle BAC = 55^{\circ}$, $\angle$CBD = 5^{\circ}$, and$\angle BEA = 180 - \angle EBA - \angle BAE = 60^{\circ}$. Draw$E'$such that$EE'B = 60^{\circ}$and so that$E'$is on$\overline{AE}$, and draw$E$such that$\angle EEC = 60^{\circ}$and$E$is on$\overline{DE}$. It follows that$\triangle BEE'$and$\triangle CEE$are both equilateral. Also, it is easy to see that$\triangle BEC \cong \triangle DEC$and$\triangle BCE \cong \triangle BAE'$by construction, so that$DE = BE = EE'$and$EE = CE = E'A$. Thus,$AE = AE' + E'E = EE + DE = DE$, so$\triangle ADE$is isosceles. Since$\angle AED = 120^{\circ}$, then$\angle DAC = \frac{180 - 120}{2} = 30^{\circ}$, and$\angle BAD = 30 + 55 = 85^{\circ}$. [asy] import graph; size(6.73cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.237,xmax=2.492,ymin=-0.16,ymax=1.947;  pen evefev=rgb(0.898,0.937,0.898), qqwuqq=rgb(0,0.392,0);  filldraw(arc((1,0),0.141,115,175)--(1,0)--cycle,evefev,qqwuqq); filldraw(arc((0.658,0.94),0.051,175,235)--(0.658,0.94)--cycle,evefev,qqwuqq); draw((0,0)--(1,0)); draw((1,0)--(0.658,0.94)); draw((0.658,0.94)--(0.158,1.806)); draw((0.158,1.806)--(0,0)); draw((0,0)--(0.658,0.94)); draw((0.158,1.806)--(1,0)); draw((0.058,0.082)--(1,0)); draw((0.558,0.948)--(0.658,0.94));  dot((0,0),ds); label("$A$",(-0.096,0.005),NE*lsf); dot((1,0),ds); label("$B$",(1.117,0.028),NE*lsf); dot((0.658,0.94),ds); label("$C$",(0.727,0.996),NE*lsf); dot((0.158,1.806),ds); label("$D$",(0.187,1.914),NE*lsf); dot((0.6,0.857),ds); label("$E$",(0.479,0.825),NE*lsf); dot((0.058,0.082),ds); label("$E'$",(0.1,0.23),NE*lsf); label("$60^\circ$",(0.767,0.091),NE*lsf,qqwuqq); dot((0.558,0.948),ds); label("$E''$",(0.423,0.957),NE*lsf); label("$60^\circ$",(0.761,0.886),NE*lsf,qqwuqq);  clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  [/asy]

See also

2008 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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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