1996 AHSME Problems/Problem 28

Problem

On a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$ , $B$ , and $C$ are adjacent to vertex $D$ . The perpendicular distance from $D$ to the plane containing $A$ , $B$ , and $C$ is closest to

$[asy] size(120); import three; currentprojection=orthographic(1, 4/5, 1/3); draw(box(O, (4,4,3))); triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); draw(A--B--C--cycle, linewidth(0.9)); label("$A$", A, NE); label("$B$", B, NW); label("$C$", C, S); label("$D$", D, E); label("$4$", (4,2,0), SW); label("$4$", (2,4,0), SE); label("$3$", (0, 4, 1.5), E); [/asy]$

$\text{(A)}\ 1.6\qquad\text{(B)}\ 1.9\qquad\text{(C)}\ 2.1\qquad\text{(D)}\ 2.7\qquad\text{(E)}\ 2.9$

Solution 1

By placing the cube in a coordinate system such that $D$ is at the origin, $A(0,0,3)$ , $B(4,0,0)$ , and $C(0,4,0)$ , we find that the equation of plane $ABC$ is:

$\frac{x}{4} + \frac{y}{4} + \frac{z}{3} = 1,$ so $3x + 3y + 4z - 12 = 0.$ The equation for the distance of a point $(a,b,c)$ to a plane $Ax + By + Cz + D = 0$ is given by:

$\frac{Aa + Bb + Cc + D}{\sqrt{A^2 + B^2 + C^2}}.$

Note that the capital letters are coefficients, while the lower case is the point itself. Thus, the distance from the origin (where $a=b=c=0$ ) to the plane is given by:

$\frac{D}{\sqrt{A^2 + B^2 + C^2}} = \frac{12}{\sqrt{9 + 9 + 16}} = \frac{12}{\sqrt{34}}.$

Since $\sqrt{34} < 6$ , this number should be just a little over $2$ , and the correct answer is $\boxed{\text{(C)}}$ .

Note that the equation above for the distance from a point to a plane is a 3D analogue of the 2D case of the distance formula, where you take the distance from a point to a line. In the 2D case, both $c$ and $C$ are set equal to $0$ .

Solution 2

Let $x$ be the desired distance. Recall that the volume of a pyramid is given by $\frac{1}{3}\cdot h \cdot B$ , where $B$ is the area of the base and $h$ is the height. Consider pyramid $ABCD$ . Letting $ABC$ be the base, the volume of $ABCD$ is given by $\frac{1}{3}\cdot x \cdot[ABC]$ , but if we let $BCD$ be the base, the volume is given by $\frac{1}{3}\cdot[BCD]\cdotAD=\frac{1}{3}\cdot(\frac{1}{2}\cdot4\cdot4)\cdot3=8$ (Error compiling LaTeX. Unknown error_msg). Clearly, these two volumes must be equal, so we get the equation $\frac{1}{3}\cdot x \cdot[ABC]=8$ . Thus, to find $x$ , we just need to find $[ABC]$ .

By the Pythagorean Theorem, $AB=\sqrt{AD^2+DB^2}=5$ , $AC=\sqrt{AD^2+DC^2}=5$ , $BC=\sqrt{BD^2+DC^2}=4\sqrt{2}$ .

The altitude to $BC$ in triangle $ABC$ has length $\sqrt{AC^2-\frac{BC}{2}^2}=\sqrt{17}$ , so $[ABC]=\frac{1}{2}\cdot 4\sqrt{2} \cdot \sqrt{17} = 2\sqrt{34}$ . Then $x=\frac{24}{[ABC]}=\frac{24}{2\sqrt{34}}=\frac{6\sqrt{34}}{17}$ or about $2.1$ . The answer is $\boxed{C}$ .

1996 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
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1996 AHSME Problems/Problem 28

Contents

Problem

Solution 1

Solution 2

See also