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Difference between revisions of "1985 AIME Problems/Problem 6"

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<math>\frac{40}{30} = \frac{124 + x}{65 + y}</math> or equivalently <math>372 + 3x = 260 + 4y</math> and so <math>4y = 3x+ 112</math>.
 
<math>\frac{40}{30} = \frac{124 + x}{65 + y}</math> or equivalently <math>372 + 3x = 260 + 4y</math> and so <math>4y = 3x+ 112</math>.
  
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Applying identical reasoning to the triangles with bases <math>\overline{CD}</math> and <math>\overline{BD}</math>, we get <math>\frac{y}{35} = \frac{x+y+84}{105}</math> so that <math>3y = x + y + 84</math> and <math>2y = x + 84</math>.  Substituting from this equation into the previous one gives <math>x = 56</math>, from which we get <math>y = 70</math> and so the area of <math>\triangle ABC</math> is <math>56 + 40 + 30 + 35 + 70 + 84 = 315</math>.
+
Applying identical reasoning to the triangles with bases <math>\overline{CD}</math> and <math>\overline{BD}</math>, we get <math>\frac{y}{35} = \frac{x+y+84}{105}</math> so that <math>3y = x + y + 84</math> and <math>2y = x + 84</math>.  Substituting from this equation into the previous one gives <math>x = 56</math>, from which we get <math>y = 70</math> and so the area of <math>\triangle ABC</math> is <math>56 + 40 + 30 + 35 + 70 + 84 = \boxed{315}</math>.
  
 
== Solution 2 ==  
 
== Solution 2 ==  

Revision as of 13:56, 21 August 2017

Problem

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.

AIME 1985 Problem 6.png

Solution

Let the interior point be $P$, let the points on $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ be $D$, $E$ and $F$, respectively. Let $x$ be the area of $\triangle APE$ and $y$ be the area of $\triangle CPD$. Note that $\triangle APF$ and $\triangle BPF$ share the same altitude from $P$, so the ratio of their areas is the same as the ratio of their bases. Similarly, $\triangle ACF$ and $\triangle BCF$ share the same altitude from $C$, so the ratio of their areas is the same as the ratio of their bases. Moreover, the two pairs of bases are actually the same, and thus in the same ratio. As a result, we have: $\frac{40}{30} = \frac{124 + x}{65 + y}$ or equivalently $372 + 3x = 260 + 4y$ and so $4y = 3x+ 112$.

Applying identical reasoning to the triangles with bases $\overline{CD}$ and $\overline{BD}$, we get $\frac{y}{35} = \frac{x+y+84}{105}$ so that $3y = x + y + 84$ and $2y = x + 84$. Substituting from this equation into the previous one gives $x = 56$, from which we get $y = 70$ and so the area of $\triangle ABC$ is $56 + 40 + 30 + 35 + 70 + 84 = \boxed{315}$.

Solution 2

This problem can be done using mass points. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight $\times$ side) = (Other weight) $\times$ (The other side), the problem yields the answer $315$

See also

1985 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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