Revision as of 07:28, 24 March 2017

Problem

A triangle has vertices $A(0,0)$ , $B(12,0)$ , and $C(8,10)$ . The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution 1

$[asy] pair A,B,C,D,X,Z,P; A=(0,0); B=(12,0); C=(8,10); X=(10,5); Z=(6,0); P=(6,3.4); fill(B--X--P--Z--cycle,lightgray); draw(A--B--C--cycle); dot(A); label("$A$",A,SW); dot(B); label("$B$",B,SE); dot(C); label("$C$",C,N); draw(X--P,dashed); draw(Z--P,dashed); dot(X); label("$X$",X,NE); dot(Z); label("$Z$",Z,S); dot(P); label("$P$",P,NW); [/asy]$

The set of all points closer to point $B$ than to point $A$ lie to the right of the perpendicular bisector of $AB$ (line $PZ$ in the diagram), and the set of all points closer to point $B$ than to point $C$ lie below the perpendicular bisector of $BC$ (line $PX$ in the diagram). Therefore, the set of points inside the triangle that are closer to $B$ than to either vertex $A$ or vertex $C$ is bounded by quadrilateral $BXPZ$ . Because $X$ is the midpoint of $BC$ and $Z$ is the midpoint of $AB$ , $X=(10,5)$ and $Z=(6,0)$ . The coordinates of point $P$ is the solution to the system of equations defined by lines $PX$ and $PZ$ . Using the point-slope form of a linear equation and the fact that the slope of the line perpendicular to a line with slope $m$ is $-\frac{1}{m}$ , the equation for line $PX$ is $y=\frac{2}{5}x+1$ and the equation for line $PZ$ is $x=6$ . The solution of this system is $P=\left(6,\frac{17}{5}\right)$ . Using the shoelace formula on quadrilateral $BXPZ$ and triangle $ABC$ , the area of quadrilateral $BXPZ$ is $\frac{109}{5}$ and the area of triangle $ABC$ is $60$ . Finally, the probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to vertex $A$ or vertex $C$ is the ratio of the area of quadrilateral $BXPZ$ to the area of $ABC$ , which is $\frac{\frac{109}{5}}{60}=\frac{109}{300}$ . The answer is $109+300=\boxed{409}$ .

Solution 2

Since we know the coordinates of all three vertices of the triangle, we can find the side lengths: $AB=12$ , $AC=2\sqrt{41}$ , and $BC=2\sqrt{29}$ . We notice that the point where the three distances are the same is the circumcenter - so we use one of the triangle area formulas to find the circumradius, since we know what the area is. $\frac{12 \cdot 2\sqrt{41} \cdot 2\sqrt{29}}{4 \cdot R}=\frac{12 \cdot 10}{2}.$ We rearrange to get $R=\frac{\sqrt{41} \cdot \sqrt{29}}{5}.$ $[asy] draw((0,0)--(12,0)--(8,10)--(0,0)); draw((6,0)--(6,3.4)--(10,5)); draw((6,3.4)--(4,5)); label("$A$", (0,0), SW); label("$B$", (12,0), SE); label("$C$", (8, 10), N); label("$P$", (6, 3.4), NNE); label("$R$", (10, 5), NE); label("$S$", (6, 0), S); label("$T$", (4, 5), NW); [/asy]$ We know that $AP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}$ , and $AS=6$ , so using the Pythagorean Theorem gives $SP=\frac{17}{5}$ . This means $[ASP]=[BSP]=\frac{17}{5} \cdot 6 \cdot \frac{1}{2} = \frac{51}{5}$ . Similarly, we know that $BP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}$ , and $BR=\sqrt{29}$ , so we get that $PR=\frac{4\sqrt{29}}{5}$ , and so $[BRP]=[CRP]=\frac{4\sqrt{29}}{5} \cdot \sqrt{29} \cdot \frac{1}{2} = \frac{58}{5}$ . Lastly, we know that $CP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}$ , and $CT=\sqrt{41}$ , so we get that $PT=\frac{2\sqrt{41}}{5}$ , and $[ATP]=[CTP]=\frac{2\sqrt{41}}{5} \cdot \sqrt{41} \cdot \frac{1}{2} = \frac{41}{5}$ . Therefore, our answer is $\frac{51+58}{2(51+58+41)}=\frac{109}{300} \rightarrow \boxed{409}$ .

@@ Line 2: / Line 2: @@
 A triangle has vertices <math>A(0,0)</math>, <math>B(12,0)</math>, and <math>C(8,10)</math>. The probability that a randomly chosen point inside the triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
-==Solution==
+==Solution 1==
 <asy>
 pair A,B,C,D,X,Z,P;
@@ Line 25: / Line 25: @@
 The set of all points closer to point <math>B</math> than to point <math>A</math> lie to the right of the perpendicular bisector of <math>AB</math> (line <math>PZ</math> in the diagram), and the set of all points closer to point <math>B</math> than to point <math>C</math> lie below the perpendicular bisector of <math>BC</math> (line <math>PX</math> in the diagram). Therefore, the set of points inside the triangle that are closer to <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> is bounded by quadrilateral <math>BXPZ</math>. Because <math>X</math> is the midpoint of <math>BC</math> and <math>Z</math> is the midpoint of <math>AB</math>, <math>X=(10,5)</math> and <math>Z=(6,0)</math>. The coordinates of point <math>P</math> is the solution to the system of equations defined by lines <math>PX</math> and <math>PZ</math>. Using the point-slope form of a linear equation and the fact that the slope of the line perpendicular to a line with slope <math>m</math> is <math>-\frac{1}{m}</math>, the equation for line <math>PX</math> is <math>y=\frac{2}{5}x+1</math> and the equation for line <math>PZ</math> is <math>x=6</math>. The solution of this system is <math>P=\left(6,\frac{17}{5}\right)</math>. Using the shoelace formula on quadrilateral <math>BXPZ</math> and triangle <math>ABC</math>, the area of quadrilateral <math>BXPZ</math> is <math>\frac{109}{5}</math> and the area of triangle <math>ABC</math> is <math>60</math>. Finally, the probability that a randomly chosen point inside the triangle is closer to vertex <math>B</math> than to vertex <math>A</math> or vertex <math>C</math> is the ratio of the area of quadrilateral <math>BXPZ</math> to the area of <math>ABC</math>, which is <math>\frac{\frac{109}{5}}{60}=\frac{109}{300}</math>. The answer is <math>109+300=\boxed{409}</math>.
+==Solution 2==
+Since we know the coordinates of all three vertices of the triangle, we can find the side lengths: <math>AB=12</math>, <math>AC=2\sqrt{41}</math>, and <math>BC=2\sqrt{29}</math>. We notice that the point where the three distances are the same is the circumcenter - so we use one of the triangle area formulas to find the circumradius, since we know what the area is.
+<cmath>\frac{12 \cdot 2\sqrt{41} \cdot 2\sqrt{29}}{4 \cdot R}=\frac{12 \cdot 10}{2}.</cmath>
+We rearrange to get
+<cmath>R=\frac{\sqrt{41} \cdot \sqrt{29}}{5}.</cmath>
+<asy>
+draw((0,0)--(12,0)--(8,10)--(0,0));
+draw((6,0)--(6,3.4)--(10,5));
+draw((6,3.4)--(4,5));
+label("$A$", (0,0), SW);
+label("$B$", (12,0), SE);
+label("$C$", (8, 10), N);
+label("$P$", (6, 3.4), NNE);
+label("$R$", (10, 5), NE);
+label("$S$", (6, 0), S);
+label("$T$", (4, 5), NW);
+</asy>
+We know that <math>AP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}</math>, and <math>AS=6</math>, so using the Pythagorean Theorem gives <math>SP=\frac{17}{5}</math>. This means <math>[ASP]=[BSP]=\frac{17}{5} \cdot 6 \cdot \frac{1}{2} = \frac{51}{5}</math>.
+Similarly, we know that <math>BP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}</math>, and <math>BR=\sqrt{29}</math>, so we get that <math>PR=\frac{4\sqrt{29}}{5}</math>, and so <math>[BRP]=[CRP]=\frac{4\sqrt{29}}{5} \cdot \sqrt{29} \cdot \frac{1}{2} = \frac{58}{5}</math>.
+Lastly, we know that <math>CP=\frac{\sqrt{41} \cdot \sqrt{29}}{5}</math>, and <math>CT=\sqrt{41}</math>, so we get that <math>PT=\frac{2\sqrt{41}}{5}</math>, and <math>[ATP]=[CTP]=\frac{2\sqrt{41}}{5} \cdot \sqrt{41} \cdot \frac{1}{2} = \frac{41}{5}</math>.
+Therefore, our answer is <math>\frac{51+58}{2(51+58+41)}=\frac{109}{300} \rightarrow \boxed{409}</math>.
 =See Also=
 {{AIME box|year=2017|n=II|num-b=2|num-a=4}}
 {{MAA Notice}}

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Difference between revisions of "2017 AIME II Problems/Problem 3"

Revision as of 07:28, 24 March 2017

Contents

Problem

Solution 1

Solution 2

See Also