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

Latest revision as of 21:23, 4 October 2023

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 $p+q.$

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

Solution 1

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 $-5 = \frac{q - \frac{39}{2}}{p - \frac{35}{2}}$ . Cross multiply and simplify to yield that $-5p + \frac{35 \cdot 5}{2} = q - \frac{39}{2}$ , so $q = -5p + 107$ .

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) (alternatively, we could use the Shoelace Theorem). 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 $p + q = \boxed{047}$ .

Solution 2

Using the equation of the median from above, 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 $p + q = \fbox{047}$ .

Solution 3

Again, the midpoint $M$ of line segment $\overline{BC}$ is at $\left(\frac{35}{2}, \frac{39}{2}\right)$ . 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 $\frac{35}{2} - \frac{5}2 + \frac{39}{2} + \frac{25}{2} = \fbox{047}$ .

Solution 4

Plug points into the Shoelace Theorem. This will provide you with the equation $|-p - 197 + 11q| = 140$ . The find the midpoint of the line $BC$ which is $(17.5,19.5)$ . Now using this post and the given slope of the median, $-5$ , using basic algebra we can find the equation of the median which is $q = -5p + 107$ . Now that we have been given $q$ in terms of $p$ plug this equation back into $|-p - 197 + 11q| = 140$ . The result is the equation $|980 - 56p| = 140$ . Solve this equation for two possible answers $p = 15, 20$ . Plugging into $q = -5p + 107$ these inputs produce $q$ values $32$ and $7$ . Obviously $15 + 32$ is the greater sum so the answer is $47$ and we are done.

@@ Line 15: / Line 15: @@
 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{39}{2}}{p - \frac{35}{2}}</math>. Cross multiply and simplify to yield that <math>-5p + \frac{35 \cdot 5}{2} = q - \frac{39}{2}</math>, so <math>q = -5p + 107</math>.
-Use [[determinant]]s to find that the [[area]] of <math>\triangle ABC</math> is <math>\frac{1}{2}  $| \begin{matrix} p & 12 & 23 \\ q & 19 & 20 \\ 1 & 1 & 1 \end{matrix} |$  = 70</math> (note that there is a missing [[absolute value]]; we will assume that the other solution for the triangle will give a smaller value of <math>p+q</math>, which is provable by following these steps over again) (alternatively, we could use the [[Shoelace Theorem]]). We can calculate this determinant to become <math>140 =  $| \begin{matrix} 12 & 23 \\ 19 & 20 \end{matrix} |$  -  $| \begin{matrix} p & q \\ 23 & 20 \end{matrix} |$  +  $| \begin{matrix} p & q \\ 12 & 19 \end{matrix} |$ </math> <math>\Longrightarrow 140 = 240 - 437 - 20p + 23q + 19p - 12q</math> <math>= -197 - p + 11q</math>. Thus, <math>q = \frac{1}{11}p - \frac{337}{11}</math>.
+Use [[determinant]]s to find that the [[area]] of <math>\triangle ABC</math> is <math>\frac{1}{2}  $| \begin{matrix} p & 12 & 23 \\ q & 19 & 20 \\ 1 & 1 & 1 \end{matrix} |$  = 70</math> (note that there is a missing [[absolute value]]; we will assume that the other solution for the triangle will give a smaller value of <math>p+q</math>, which is provable by following these steps over again) (alternatively, we could use the [[Shoelace Theorem]]). We can calculate this determinant to become <math>140 =  $| \begin{matrix} 12 & 23 \\ 19 & 20 \end{matrix} |$  -  $| \begin{matrix} p & q \\ 23 & 20 \end{matrix} |$  +  $| \begin{matrix} p & q \\ 12 & 19 \end{matrix} |$ </math> <math>\Longrightarrow 140 = 240 - 437 - 20p + 23q + 19p - 12q</math> <math>= -197 - p + 11q</math>. Thus, <math>q = \frac{1}{11}p + \frac{337}{11}</math>.
-Setting this equation equal to the equation of the median, we get that <math>\frac{1}{11}p - \frac{337}{11} = -5p + 107</math>, so <math>\frac{56}{11}p = \frac{107 \cdot 11 + 337}{11}</math>. Solving produces that <math>p = 15</math>. [[Substitution|Substituting]] backwards yields that <math>q = 32</math>; the solution is <math>p + q = \boxed{047}</math>.
+Setting this equation equal to the equation of the median, we get that <math>\frac{1}{11}p + \frac{337}{11} = -5p + 107</math>, so <math>\frac{56}{11}p = \frac{107 \cdot 11 - 337}{11}</math>. Solving produces that <math>p = 15</math>. [[Substitution|Substituting]] backwards yields that <math>q = 32</math>; the solution is <math>p + q = \boxed{047}</math>.
 == Solution 2 ==

2005 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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