Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1999 AIME Problems/Problem 15"
m (Proofreading)
Line 4: Line 4:
 
== Solution ==
 
== Solution ==
 
<center>[[Image:AIME_1999_Solution_15_1.png]][[Image:AIME_1999_Solution_15_2.png]]</center>
 
<center>[[Image:AIME_1999_Solution_15_1.png]][[Image:AIME_1999_Solution_15_2.png]]</center>
−
Let <math>D</math>, <math>E</math>, <math>F</math> be the foots of the altitudes of sides <math>BC</math>, <math>CA</math>, <math>AB</math>, respectively, of <math>\triangle ABC</math>.
+
Let <math>D</math>, <math>E</math>, <math>F</math> be the feet of the altitudes to sides <math>BC</math>, <math>CA</math>, <math>AB</math>, respectively, of <math>\triangle ABC</math>.
 
The base of the [[tetrahedron]] is the [[orthocenter]] <math>O</math> of the large triangle, so we just need to find that, then it's easy from there.
 
The base of the [[tetrahedron]] is the [[orthocenter]] <math>O</math> of the large triangle, so we just need to find that, then it's easy from there.
  
−
To find the coordinates of <math>O</math>, we wish to find the intersection point of altitudes <math>BE</math> and <math>AD</math>. The equation of <math>BE</math> is just <math>x=16</math> . <math>AD</math> is perpendicular to the line <math>BC</math>, so the slope of <math>AD</math> is equal to the negative reciprocal of the slope of <math>BC</math>. <math>BC</math> has slope <math>\frac{24-0}{16-34}=-\frac{4}{3}</math>, therefore <math>y=\dfrac{3}{4} x</math>. These two equations intersect at <math>(16,12)</math>, so that's the base of the height of the tetrahedron.  
+
To find the coordinates of <math>O</math>, we need to find the intersection point of altitudes <math>BE</math> and <math>AD</math>. The equation of <math>BE</math> is simply <math>x=16</math>. <math>AD</math> is [[perpendicular]] to line <math>BC</math>, so the slope of <math>AD</math> is equal to the negative reciprocal of the slope of <math>BC</math>. <math>BC</math> has slope <math>\frac{24-0}{16-34}=-\frac{4}{3}</math>, therefore <math>y=\frac{3}{4} x</math>. These two lines intersect at <math>(16,12)</math>, so that's the base of the height of the tetrahedron.  
  
 
Let <math>S</math> be the foot of altitude <math>BS</math> in <math>\triangle BPQ</math>. From the pythagorean theorem, <math>h=\sqrt{BS^2-SO^2}</math>. However, since <math>S</math> and <math>O</math> are, by coincidence, the same point, <math>SO=0</math> and <math>h=12</math>.
 
Let <math>S</math> be the foot of altitude <math>BS</math> in <math>\triangle BPQ</math>. From the pythagorean theorem, <math>h=\sqrt{BS^2-SO^2}</math>. However, since <math>S</math> and <math>O</math> are, by coincidence, the same point, <math>SO=0</math> and <math>h=12</math>.
  
−
And the area of the base is 104, so the volume is
+
The area of the base is <math>104</math>, so the volume is <math>\frac{104*12}{3}=\boxed{408}</math>.
−
 
 
−
<math>\dfrac{104*12}{3}=\boxed{408}</math>
 
  
 
== See also ==
 
== See also ==
−
{{AIME box|year=1999|num-b=14|after=Final Question}}
+
{{AIME box|year=1999|num-b=14|after=Last Question}}

Revision as of 12:02, 26 April 2008

Problem

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

Solution

AIME 1999 Solution 15 1.pngAIME 1999 Solution 15 2.png

Let $D$, $E$, $F$ be the feet of the altitudes to sides $BC$, $CA$, $AB$, respectively, of $\triangle ABC$. The base of the tetrahedron is the orthocenter $O$ of the large triangle, so we just need to find that, then it's easy from there.

To find the coordinates of $O$, we need to find the intersection point of altitudes $BE$ and $AD$. The equation of $BE$ is simply $x=16$. $AD$ is perpendicular to line $BC$, so the slope of $AD$ is equal to the negative reciprocal of the slope of $BC$. $BC$ has slope $\frac{24-0}{16-34}=-\frac{4}{3}$, therefore $y=\frac{3}{4} x$. These two lines intersect at $(16,12)$, so that's the base of the height of the tetrahedron.

Let $S$ be the foot of altitude $BS$ in $\triangle BPQ$. From the pythagorean theorem, $h=\sqrt{BS^2-SO^2}$. However, since $S$ and $O$ are, by coincidence, the same point, $SO=0$ and $h=12$.

The area of the base is $104$, so the volume is $\frac{104*12}{3}=\boxed{408}$.

See also

1999 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1999_AIME_Problems/Problem_15&oldid=25550"