Difference between revisions of "2003 AMC 12A Problems/Problem 16"

Revision as of 09:06, 23 July 2015

Problem

A point P is chosen at random in the interior of equilateral triangle $ABC$ . What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$ ?

$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{2}{3}$

Solution

Solution 1

After we pick point $P$ , we realize that $ABC$ is symmetric for this purpose, and so the probability that $ACP$ is the greatest area, or $ABP$ or $BCP$ , are all the same. Since they add to $1$ , the probability that $ACP$ has the greatest area is $\boxed{\mathrm{(C)}\ \dfrac{1}{3}}$

Solution 2

We will use an approach of geometric probability to solve this problem. Let us take point P, and draw the perpendiculars to BC, CA, and AB, and call the feet of these perpendiculars D, E, and F respectively. The area of triangle ACP is simply $\frac{1}{2} * AC * PF$ . Similarly we can find the area of triangles BCP and ABP. If we add these up and realize that it equals the area of the entire triangle, we see that no matter where we choose P, PD + PE + PF = the height of the triangle. Setting the area of triangle ABP greater than ACP and BCP, we want PF to be the largest of PF, PD, and PE. We then realize that PF = PD = PE when P is the incenter of ABC. Let us call the incenter of the triangle Q. If we want PF to be the largest of the three, by testing points we realize that P must be in the interior of quadrilateral QDCE. So our probability (using geometric probability) is the area of QDCE divided by the area of ABC. We will now show that the three quadrilaterals, QDCE, QEAF, and QFBD are congruent. As the definition of point Q yields, QF = QD = QE. Since ABC is equilateral, Q is also the circumcenter of ABC, so QA = QB = QC. By the Pythagorean Theorem, BD = DC = CE = EA = AF = FB. Also, angles BDQ, BFQ, CEQ, CDQ, AFQ, and AEQ are all equal to 90 degrees. Angles DBF, FAE, ECD are all equal to 60 degrees, so it is now clear that QDCE, QEAF, QFBD are all congruent. Summing up these areas gives us the area of ABC. QDCE contributes to a third of that area so $\frac{[QDCE]}{[ABC]}=\frac{1}{3}$ (C).

@@ Line 12: / Line 12: @@
 ===Solution 2===
-We will use an approach of geometric probability to solve this problem. Let us take point P, and draw the perpendiculars to AB, BC, and AC, and call the feet of these perpendiculars D, E, and F respectively. The area of triangle ACP is simply 1/2 * AC * PF. Similarly we can find the area of triangles BCP and ABP. If we add these up and realize that it equals the area of the entire triangle, we see that no matter where we choose P, PD + PE + PF = the height of the triangle. Setting the area of triangle ACP greater than ABP and BCP, we want PF to be the largest of PF, PD, and PE. We then realize that PF = PD = PE when P is the orthocenter of ABC. Let us call the orthocenter of the triangle Q. If we want PF to be the largest of the three, by testing points we realize that P must be in the interior of quadrilateral QFCE. So our probability (using geometric probability) is the area of QFCE divided by the area of ABC. We will now show that the three quadrilaterals, QFCE, QEBD, and QDAF are congruent. As the definition of point Q yields, QF = QD = QE. Since ABC is equilateral, Q is also the circumcenter of ABC, so QA = QB = QC. Using the Pythagorean theorem, BD = DA = AF = FC = CE = EB. Also, angles BDQ, BEQ, CEQ, CFQ, AFQ, and ADQ are all equal to 90 degrees by the definition of an altitude. Also, angles DBE, FCE, DAF are all equal to 60 degrees as equilateral triangles are also equiangular. It is now clear that QFCE, QFAD, QEBD are all congruent. Summing up these areas gives us the area of ABC. QFCE contributes to a third of that area, as they are all congruent, so the ratio of the areas of QFCE to ABC is 1/3 (C).
+We will use an approach of geometric probability to solve this problem. Let us take point P, and draw the perpendiculars to BC, CA, and AB, and call the feet of these perpendiculars D, E, and F respectively. The area of triangle ACP is simply <math>\frac{1}{2} * AC * PF</math>. Similarly we can find the area of triangles BCP and ABP. If we add these up and realize that it equals the area of the entire triangle, we see that no matter where we choose P, PD + PE + PF = the height of the triangle. Setting the area of triangle ABP greater than ACP and BCP, we want PF to be the largest of PF, PD, and PE. We then realize that PF = PD = PE when P is the incenter of ABC. Let us call the incenter of the triangle Q. If we want PF to be the largest of the three, by testing points we realize that P must be in the interior of quadrilateral QDCE. So our probability (using geometric probability) is the area of QDCE divided by the area of ABC. We will now show that the three quadrilaterals, QDCE, QEAF, and QFBD are congruent. As the definition of point Q yields, QF = QD = QE. Since ABC is equilateral, Q is also the circumcenter of ABC, so QA = QB = QC. By the Pythagorean Theorem, BD = DC = CE = EA = AF = FB. Also, angles BDQ, BFQ, CEQ, CDQ, AFQ, and AEQ are all equal to 90 degrees. Angles DBF, FAE, ECD are all equal to 60 degrees, so it is now clear that QDCE, QEAF, QFBD are all congruent. Summing up these areas gives us the area of ABC. QDCE contributes to a third of that area so <math>\frac{[QDCE]}{[ABC]}=\frac{1}{3}</math> (C).
 ==See Also==
 {{AMC12 box|year=2003|ab=A|num-b=15|num-a=17}}
 {{MAA Notice}}

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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