Difference between revisions of "1999 AIME Problems/Problem 15"

Latest revision as of 12:26, 3 September 2024

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 1

$[asy]defaultpen(fontsize(9)+linewidth(0.63)); pair A=(0,0), B=(16,24), C=(34,0), P=(8,12), Q=(25,12), R=(17,0); draw(A--B--C--A);draw(P--Q--R--P); draw(A--foot(A,B,C));draw(B--foot(B,A,C));draw(C--foot(C,A,B)); label("$A$",A,SW);label("$B$",B,NW);label("$C$",C,SE); label("$D$",foot(A,B,C),NE);label("$E$",foot(B,A,C),SW);label("$F$",foot(C,A,B),NW);label("$P$",P,NW);label("$Q$",Q,NE);label("$R$",R,SE);[/asy]$ $[asy]import three; defaultpen(linewidth(0.6)); currentprojection=orthographic(1/2,-1,1/2); triple A=(0,0,0), B=(16,24,0), C=(34,0,0), P=(8,12,0), Q=(25,12,0), R=(17,0,0), S=(16,12,12); draw(A--B--C--A); draw(P--Q--R--P); draw(S--P..S--Q..S--R); draw(S--(16,12,0)); [/asy]$

As shown in the image above, let $D$ , $E$ , and $F$ be the midpoints of $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ , respectively. Suppose $P$ is the apex of the tetrahedron, and let $O$ be the foot of the altitude from $P$ to $\triangle ABC$ . The crux of this problem is the following lemma.

Lemma: The point $O$ is the orthocenter of $\triangle ABC$ .

Proof. Observe that $OF^2 - OE^2 = PF^2 - PE^2 = AF^2 - AE^2;$ the first equality follows by the Pythagorean Theorem, while the second follows from $AF = FP$ and $AE = EP$ . Thus, by the Perpendicularity Lemma, $AO$ is perpendicular to $FE$ and hence $BC$ . Analogously, $O$ lies on the $B$ -altitude and $C$ -altitude of $\triangle ABC$ , and so $O$ is, indeed, the orthocenter of $\triangle ABC$ .

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 $102$ , so the volume is $\frac{102*12}{3}=\boxed{408}$ .~Shen Kislay Kai

Solution 2

Consider the diagram provided in the previous solution. We first note that the medial triangle has coordinates $(17, 0, 0)$ , $(8, 12, 0)$ , and $(25, 12, 0)$ . We can compute the area of this triangle as $102$ . Suppose $(x, y, z)$ are the coordinates of the vertex of the resulting pyramid. Call this point $V$ . Clearly, the height of the pyramid is $z$ . The desired volume is thus $\frac{102z}{3} = 34z$ .

We note that when folding the triangle to form the pyramid, some side lengths must stay the same. In particular, $VR = RA$ , $VP = PB$ , and $VQ = QC$ . We then use distance formula to find the distances from $V$ to each of the vertices of the medial triangle. We thus arrive at a fairly simple system of equations, yielding $z = 12$ . The desired volume is thus $34 \times 12 = \boxed{408}$ .~Shen Kislay Kai

Solution 3

The formed tetrahedron has pairwise parallel planar and oppositely equal length ( $4\sqrt{13},15,17$ ) edges and can be inscribed in a parallelepiped (rectangular box) with the six tetrahedral edges as non-intersecting diagonals of the box faces. Let the edge lengths of the parallelepiped be $p,q,r$ and solve (by Pythagoras)

$p^2+q^2=4^2\cdot{13}$

$q^2+r^2=15^2$

$r^2+p^2=17^2$

to find that $(p^2,q^2,r^2)=(153,136,72)=(3^2\cdot{17},2^3\cdot{17},2^3\cdot{3^2}).$

Use the fact that the ratio of volumes between an inscribed tetrahedron and its circumscribing parallelepiped is $\tfrac{1}{3}$ and then the volume is

$\tfrac{1}{3}pqr=\tfrac{1}{3}\sqrt{2^6\cdot{3^4}\cdot{17^2}}=\boxed{408}$

Solution by D. Adrian Tanner

Solution 4

Let $A = (0,0), B = (16, 24), C = (34,0).$ Then define $D,E,F$ as the midpoints of $BC, AC, AB$ . By Pythagorean theorem, $EF = \frac{1}{2} BC = 15, DE = \frac{1}{2}AB = 4 \sqrt{13}, DF = \frac{1}{2} AC = 17.$ Then let $P$ be the point in space which is the vertex of the tetrahedron with base $DEF$ .

Note that $\triangle DEP \cong \triangle EDF$ . Create point $F'$ on the plane of $DEF$ such that $\triangle DEP \cong \triangle DEF'$ (i.e by reflecting $F$ over the perpendicular bisector of $DE$ ). Project $F, P$ onto $DE$ as $X, Y$ . Note by the definition of $F'$ then $\angle PYF'$ is the dihedral angle between planes $DEP, DEF$ .

Now see that by Heron's, $[DEP] = [DEF] = \sqrt{(16 + 2 \sqrt{13})(16 - 2 \sqrt{13})(1 + 2 \sqrt{13})(-1 + 2 \sqrt{13})} = 102.$ So $PY$ , the hypotenuse $DEP$ has length $\frac{102 \cdot 2}{4 \sqrt{13}} = \frac{51}{\sqrt{13}}$ . Similarly $F'Y = \frac{51}{\sqrt{13}}.$ Further from Pythagoras $DY = \sqrt{DP^2 - PY^2} = \frac{18}{\sqrt{13}}.$ Symmetrically $EX = \frac{18}{\sqrt{13}}.$ Therefore $XY = DE - DY - EX = \frac{16}{\sqrt{13}}.$

By Law of Cosines on $\triangle PYF'$ , $\begin{align*} PF'^2 &= PY^2 + F'Y^2 - 2 \cdot PY \cdot F'Y \cos{\angle PYF'} \\ PF^2 - XY^2 &= 2 (\frac{51}{\sqrt{13}})^2 \cos{\angle PYF'} \\ (4\sqrt{13})^2 - (\frac{16}{\sqrt{13}})^2 &= 2 (\frac{51}{\sqrt{13}})^2 \cos{\angle PYF'} \\ \cos{\angle PYF'} &= \frac{9}{17} \\ \sin{\angle PYF'} &= \frac{4 \sqrt{13}}{17}. \end{align*}.$

Therefore the altitude of the tetrahedron from vertex $P$ to base $DEF$ is $PY \sin{\angle PYF'} = \frac{51}{\sqrt{13}} \frac{4 \sqrt{13}}{17} = 12.$ So the area is $\frac{1}{3}bh = \frac{1}{3} 12 \cdot 102 = \boxed{408}.$

~ Aaryabhatta1

Solution 5 (Formula Abuse)

The Pyramid is a disphenoid, because opposite sides have the same length. The volume of a disphenoid is given by

$V = \sqrt{\frac{(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}.$

Using the Pythagorean theorem, the side lengths of the smaller triangle are $15$, $4\sqrt{13}$, and $17$. Plugging in, we get

$V = \sqrt{166464} = \boxed{408}.$

~~Disphenoid_lover

@@ Line 2: / Line 2: @@
 Consider the paper triangle whose vertices are <math>(0,0), (34,0),</math> and <math>(16,24).</math>  The vertices of its midpoint triangle are the [[midpoint]]s 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 ==
+== Solution 1 ==
 <center><asy>defaultpen(fontsize(9)+linewidth(0.63)); pair A=(0,0), B=(16,24), C=(34,0), P=(8,12), Q=(25,12), R=(17,0); draw(A--B--C--A);draw(P--Q--R--P); draw(A--foot(A,B,C));draw(B--foot(B,A,C));draw(C--foot(C,A,B)); label("\(A\)",A,SW);label("\(B\)",B,NW);label("\(C\)",C,SE); label("\(D\)",foot(A,B,C),NE);label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><asy>import three; defaultpen(linewidth(0.6));
 currentprojection=orthographic(1/2,-1,1/2); triple A=(0,0,0), B=(16,24,0), C=(34,0,0), P=(8,12,0), Q=(25,12,0), R=(17,0,0), S=(16,12,12); draw(A--B--C--A); draw(P--Q--R--P); draw(S--P..S--Q..S--R); draw(S--(16,12,0)); </asy></center><!-- Asymptote renderings of Image:AIME_1999_Solution_15_1.png, Image:AIME_1999_Solution_15_2.png, by Minsoens -->
-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>.
+As shown in the image above, let <math>D</math>, <math>E</math>, and <math>F</math> be the midpoints of <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively.  Suppose <math>P</math> is the apex of the tetrahedron, and let <math>O</math> be the foot of the altitude from <math>P</math> to <math>\triangle ABC</math>.  The crux of this problem is the following lemma.
-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.
+<b>Lemma:</b> The point <math>O</math> is the orthocenter of <math>\triangle ABC</math>.
+<i>Proof.</i> Observe that <cmath>OF^2 - OE^2 = PF^2 - PE^2 = AF^2 - AE^2;</cmath> the first equality follows by the Pythagorean Theorem, while the second follows from <math>AF = FP</math> and <math>AE = EP</math>.  Thus, by the Perpendicularity Lemma, <math>AO</math> is perpendicular to <math>FE</math> and hence <math>BC</math>.  Analogously, <math>O</math> lies on the <math>B</math>-altitude and <math>C</math>-altitude of <math>\triangle ABC</math>, and so <math>O</math> is, indeed, the orthocenter of <math>\triangle ABC</math>.
 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.
@@ Line 12: / Line 15: @@
 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>.
-The area of the base is <math>104</math>, so the volume is <math>\frac{104*12}{3}=\boxed{408}</math>.
+The area of the base is <math>102</math>, so the volume is <math>\frac{102*12}{3}=\boxed{408}</math>.~Shen Kislay Kai
+== Solution 2 ==
+Consider the diagram provided in the previous solution. We first note that the medial triangle has coordinates <math>(17, 0, 0)</math>, <math>(8, 12, 0)</math>, and <math>(25, 12, 0)</math>. We can compute the area of this triangle as <math>102</math>. Suppose <math>(x, y, z)</math> are the coordinates of the vertex of the resulting pyramid. Call this point <math>V</math>. Clearly, the height of the pyramid is <math>z</math>. The desired volume is thus <math>\frac{102z}{3} = 34z</math>.
+We note that when folding the triangle to form the pyramid, some side lengths must stay the same. In particular, <math>VR = RA</math>, <math>VP = PB</math>, and <math>VQ = QC</math>. We then use distance formula to find the distances from <math>V</math> to each of the vertices of the medial triangle. We thus arrive at a fairly simple system of equations, yielding <math>z = 12</math>. The desired volume is thus <math>34 \times 12 = \boxed{408}</math>.~Shen Kislay Kai
+== Solution 3 ==
+The formed tetrahedron has pairwise parallel planar and oppositely equal length (<math>4\sqrt{13},15,17</math>) edges and can be inscribed in a parallelepiped (rectangular box) with the six tetrahedral edges as non-intersecting diagonals of the box faces.  Let the edge lengths of the parallelepiped be <math>p,q,r</math> and solve (by Pythagoras)
+<math>p^2+q^2=4^2\cdot{13}</math>
+<math>q^2+r^2=15^2</math>
+<math>r^2+p^2=17^2</math>
+to find that <math>(p^2,q^2,r^2)=(153,136,72)=(3^2\cdot{17},2^3\cdot{17},2^3\cdot{3^2}).</math>
+Use the fact that the ratio of volumes between an inscribed tetrahedron and its circumscribing parallelepiped is <math>\tfrac{1}{3}</math> and then the volume is
+<math>\tfrac{1}{3}pqr=\tfrac{1}{3}\sqrt{2^6\cdot{3^4}\cdot{17^2}}=\boxed{408}</math>
+Solution by D. Adrian Tanner
+== Solution 4 ==
+Let <math>A = (0,0), B = (16, 24), C = (34,0).</math> Then define <math>D,E,F</math> as the midpoints of <math>BC, AC, AB</math>. By Pythagorean theorem, <math>EF = \frac{1}{2} BC = 15, DE = \frac{1}{2}AB = 4 \sqrt{13}, DF = \frac{1}{2} AC = 17.</math> Then let <math>P</math> be the point in space which is the vertex of the tetrahedron with base <math>DEF</math>.
+Note that <math>\triangle DEP \cong \triangle EDF</math>. Create point <math>F'</math> on the plane of <math>DEF</math> such that <math>\triangle DEP \cong \triangle DEF'</math> (i.e by reflecting <math>F</math> over the perpendicular bisector of <math>DE</math>). Project <math>F, P</math> onto <math>DE</math> as <math>X, Y</math>. Note by the definition of <math>F'</math> then <math>\angle PYF'</math> is the dihedral angle between planes <math>DEP, DEF</math>.
+Now see that by Heron's, <cmath>[DEP] = [DEF] = \sqrt{(16 + 2 \sqrt{13})(16 - 2 \sqrt{13})(1 + 2 \sqrt{13})(-1 + 2 \sqrt{13})} = 102.</cmath>
+So <math>PY</math>, the hypotenuse <math>DEP</math> has length <math>\frac{102 \cdot 2}{4 \sqrt{13}} = \frac{51}{\sqrt{13}}</math>. Similarly <math>F'Y = \frac{51}{\sqrt{13}}.</math> Further from Pythagoras <math>DY = \sqrt{DP^2 - PY^2} = \frac{18}{\sqrt{13}}.</math> Symmetrically <math>EX = \frac{18}{\sqrt{13}}.</math> Therefore <math>XY = DE - DY - EX = \frac{16}{\sqrt{13}}.</math>
+By Law of Cosines on <math>\triangle PYF'</math>,
+<cmath>\begin{align*}
+PF'^2 &= PY^2 + F'Y^2 - 2 \cdot PY \cdot F'Y \cos{\angle PYF'} \\
+PF^2 - XY^2 &= 2 (\frac{51}{\sqrt{13}})^2 \cos{\angle PYF'} \\
+(4\sqrt{13})^2 - (\frac{16}{\sqrt{13}})^2 &= 2 (\frac{51}{\sqrt{13}})^2 \cos{\angle PYF'}  \\
+\cos{\angle PYF'} &= \frac{9}{17} \\
+\sin{\angle PYF'} &= \frac{4 \sqrt{13}}{17}.
+\end{align*}.</cmath>
+Therefore the altitude of the tetrahedron from vertex <math>P</math> to base <math>DEF</math> is <math>PY \sin{\angle PYF'} = \frac{51}{\sqrt{13}} \frac{4 \sqrt{13}}{17} = 12.</math> So the area is <math>\frac{1}{3}bh = \frac{1}{3} 12 \cdot 102 = \boxed{408}.</math>
+~ Aaryabhatta1
+== Solution 5 (Formula Abuse) ==
+The Pyramid is a disphenoid, because opposite sides have the same length. The volume of a disphenoid is given by
+<cmath>V = \sqrt{\frac{(l^{2}+m^{2}-n^{2})(l^{2}-m^{2}+n^{2})(-l^{2}+m^{2}+n^{2})}{72}}.</cmath>
+Using the Pythagorean theorem, the side lengths of the smaller triangle are \(15\), \(4\sqrt{13}\), and \(17\). Plugging in, we get
+<cmath>V = \sqrt{166464} = \boxed{408}.</cmath>
+~~Disphenoid_lover
 == See also ==
@@ Line 18: / Line 80: @@
 [[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

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