Difference between revisions of "2005 AIME I Problems/Problem 10"

m (Solution: image needed)
m
Line 4: Line 4:
 
__TOC__
 
__TOC__
 
== Solution ==
 
== Solution ==
{{image}}
+
<center><asy>defaultpen(fontsize(8));
The [[midpoint]] <math>M</math> of [[line segment]] <math>\overline{BC}</math> is <math>\left(\frac{35}{2}, \frac{39}{2}\right)</math>. The equation of the median can be found by <math>-5 = \frac{q - \frac{35}{2}}{p - \frac{39}{2}}</math>. Cross multiply and simplify to yield that <math>\displaystyle -5p + \frac{39 \cdot 5}{2} = q - \frac{35}{2}</math>, so <math>q = -5p + 107</math>.
+
size(170);
 +
pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22);
 +
draw(A--B--C--A);draw(A--M);draw(B--P--C);
 +
label("A (p,q)",A,(1,1));label("B (12,19)",B,(-1,-1));label("C (23,20)",C,(1,-1));label("M",M,(0.2,-1));
 +
label("(17,22)",P,(1,1));
 +
dot(A^^B^^C^^M^^P);</asy></center>
 +
 
 +
The [[midpoint]] <math>M</math> of [[line segment]] <math>\overline{BC}</math> is <math>\left(\frac{35}{2}, \frac{39}{2}\right)</math>. The equation of the median can be found by <math>-5 = \frac{q - \frac{35}{2}}{p - \frac{39}{2}}</math>. Cross multiply and simplify to yield that <math>-5p + \frac{39 \cdot 5}{2} = q - \frac{35}{2}</math>, so <math>q = -5p + 107</math>.
  
 
=== Solution 1 ===
 
=== Solution 1 ===
Line 13: Line 20:
  
 
=== Solution 2 ===
 
=== Solution 2 ===
Using the equation of the median, we can write the [[coordinate]]s of <math>A</math> as <math>(p,\ -5p + 107)</math>. The equation of <math>\overline{BC}</math> is <math>\frac{20 - 19}{23 - 12} = \frac{y - 19}{x - 12}</math>, so <math>x - 12 = 11y - 209</math>. In [[general form]], the line is <math>x - 11y + 197 = 0</math>. Use the equation for the distance between a line and point to find the distance between <math>A</math> and <math>BC</math> (which is the height of <math>\triangle ABC</math>): <math>\frac{|1(p) - 11(-5p + 107) + 197|}{1^2 + 11^2} = \frac{|56p - 980|}{\sqrt{122}}</math>. Now we need the length of <math>BC</math>, which is <math>\sqrt{(23 - 12)^2 + (20 - 19)^2} = \sqrt{122}</math>. The area of <math>\triangle ABC</math> is <math>70 = \frac{1}{2}bh = \frac{1}{2}(\frac{|56p - 980|}{\sqrt{122}}) \cdot \sqrt{122}</math>. Thus, <math>|28p - 490| = 70</math>, and <math>p = 15,\ 20</math>. We are looking for <math>p + q = -4p + 107 = 47,\ 27</math>. The maximum possible value of <math>p + q = 47</math>.  
+
Using the equation of the median, we can write the [[coordinate]]s of <math>A</math> as <math>(p,\ -5p + 107)</math>. The equation of <math>\overline{BC}</math> is <math>\frac{20 - 19}{23 - 12} = \frac{y - 19}{x - 12}</math>, so <math>x - 12 = 11y - 209</math>. In [[general form]], the line is <math>x - 11y + 197 = 0</math>. Use the equation for the distance between a line and point to find the distance between <math>A</math> and <math>BC</math> (which is the height of <math>\triangle ABC</math>): <math>\frac{|1(p) - 11(-5p + 107) + 197|}{1^2 + 11^2} = \frac{|56p - 980|}{\sqrt{122}}</math>. Now we need the length of <math>BC</math>, which is <math>\sqrt{(23 - 12)^2 + (20 - 19)^2} = \sqrt{122}</math>. The area of <math>\triangle ABC</math> is <math>70 = \frac{1}{2}bh = \frac{1}{2}\left(\frac{|56p - 980|}{\sqrt{122}}\right) \cdot \sqrt{122}</math>. Thus, <math>|28p - 490| = 70</math>, and <math>p = 15,\ 20</math>. We are looking for <math>p + q = -4p + 107 = 47,\ 27</math>. The maximum possible value of <math>p + q = 47</math>.  
  
 
=== Solution 3 ===
 
=== Solution 3 ===

Revision as of 17:59, 19 November 2007

Problem

Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$

Solution

[asy]defaultpen(fontsize(8)); size(170); pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22); draw(A--B--C--A);draw(A--M);draw(B--P--C); label("A (p,q)",A,(1,1));label("B (12,19)",B,(-1,-1));label("C (23,20)",C,(1,-1));label("M",M,(0.2,-1)); label("(17,22)",P,(1,1)); dot(A^^B^^C^^M^^P);[/asy]

The midpoint $M$ of line segment $\overline{BC}$ is $\left(\frac{35}{2}, \frac{39}{2}\right)$. The equation of the median can be found by $-5 = \frac{q - \frac{35}{2}}{p - \frac{39}{2}}$. Cross multiply and simplify to yield that $-5p + \frac{39 \cdot 5}{2} = q - \frac{35}{2}$, so $q = -5p + 107$.

Solution 1

Use determinants to find that the area of $\triangle ABC$ is $\frac{1}{2} \begin{vmatrix}p & 12 & 23 \\  q & 19 & 20 \\ 1 & 1 & 1\end{vmatrix} = 70$ (note that there is a missing absolute value; we will assume that the other solution for the triangle will give a smaller value of $p+q$, which is provable by following these steps over again). We can calculate this determinant to become $140 = \begin{vmatrix} 12 & 23 \\ 19 & 20 \end{vmatrix} - \begin{vmatrix} p & q \\ 23 & 20 \end{vmatrix} + \begin{vmatrix} p & q \\ 12 & 19 \end{vmatrix} \Longrightarrow 140 = 240 - 437 - 20p + 23q + 19p - 12q = -197 - p + 11q$. Thus, $q = \frac{1}{11}p - \frac{337}{11}$.

Setting this equation equal to the equation of the median, we get that $\frac{1}{11}p - \frac{337}{11} = -5p + 107$, so $\frac{56}{11}p = \frac{107 \cdot 11 + 337}{11}$. Solving produces that $p = 15$. Substituting backwards yields that $q = 32$; the solution is $p + q = 047$.

Solution 2

Using the equation of the median, we can write the coordinates of $A$ as $(p,\ -5p + 107)$. The equation of $\overline{BC}$ is $\frac{20 - 19}{23 - 12} = \frac{y - 19}{x - 12}$, so $x - 12 = 11y - 209$. In general form, the line is $x - 11y + 197 = 0$. Use the equation for the distance between a line and point to find the distance between $A$ and $BC$ (which is the height of $\triangle ABC$): $\frac{|1(p) - 11(-5p + 107) + 197|}{1^2 + 11^2} = \frac{|56p - 980|}{\sqrt{122}}$. Now we need the length of $BC$, which is $\sqrt{(23 - 12)^2 + (20 - 19)^2} = \sqrt{122}$. The area of $\triangle ABC$ is $70 = \frac{1}{2}bh = \frac{1}{2}\left(\frac{|56p - 980|}{\sqrt{122}}\right) \cdot \sqrt{122}$. Thus, $|28p - 490| = 70$, and $p = 15,\ 20$. We are looking for $p + q = -4p + 107 = 47,\ 27$. The maximum possible value of $p + q = 47$.

Solution 3

Let $A'$ be the point $(17, 22)$, which lies along the line through $M$ of slope $-5$. The area of triangle $A'BC$ can be computed in a number of ways (one possibility: extend $A'B$ until it hits the line $y = 19$, and subtract one triangle from another), and each such calculation gives an area of 14. This is $\frac{1}{5}$ of our needed area, so we simply need the point $A$ to be 5 times as far from $M$ as $A'$ is. Thus $A = \left(\frac{35}{2}, \frac{39}{2}\right) \pm 5\left(-\frac{1}{2}, \frac{5}{2}\right)$, and the sum of coordinates will be larger if we take the positive value, so $A = \left(\frac{35}{2} - \frac{5}2, \frac{39}{2} + \frac{25}{2}\right)$ and the answer is $\frac{35}{2} - \frac{5}2 + \frac{39}{2} + \frac{25}{2} = 047$.

See also

2005 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS