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Let $A$, $B$, $C$, and $D$ be four points in space such that $$\angle BAC=\angle CAD=\angle DAB=60^\circ.$$ If $AB=1$, $AC=2$, and $AD=6$, then what is the distance between $A$ and the plane of $\triangle BCD$?


Observations: $ABC$ and $BCD$ are right, as $ABC$ is $30-60-90$ and then using Law of Cosines we han find the other side lengths for $BCD.$ I don't know how to use this to solve for the distance desired though. I have tried letting this distance be $h$ and setting up equations.

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  • $\begingroup$ One idea would be to construct these points by hand. That's not as awful as it may sound: Without loss of generality, you can pick $A=(0,0,0), B=(1,0,0), C=(1,\sqrt{3},0)$. It then remains to find $D$. $\endgroup$ Apr 2, 2021 at 20:57

5 Answers 5

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You can use the formula for tetrahedron volume (wiki),

$V = \frac{abc}{6} \sqrt{1+2 \cos \alpha \cos \beta \cos \gamma - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma}$

$\alpha, \beta, \gamma$ are angles between edges at a vertex and $a, b, c$ are lengths of edges from the vertex.

Here we know that at vertex $A$, $\angle BAC = \angle BAD = \angle CAD = 60^0$ and $AB = 1, AC = 2, AD = 6$.

So $V = \displaystyle \frac{1\cdot2\cdot6}{6} \sqrt{1 + \frac{2}{2^3} - \frac{3}{2^2}} = \sqrt2$

We also know that the volume of Tetrahedron is given by $V = \frac{1}{3} A \cdot h$ where $A$ is the area of the base and $h$ is the altitude.

As you found out, $\triangle BCD$ is right angled triangle with $BC = \sqrt3, CD = \sqrt{28}, BD = \sqrt{31}$. So, $A = \sqrt{21}$.

So altitude from $A$ to base BCD, $h = \frac{3V}{A} = \sqrt{\frac{6}{7}}$

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Let $\theta$ be the angle between the planes $ABC$ and $BCD$. Recognize the right triangles $ABC$ and $BCD$ to establish the distance equation

$$AD^2 = (AB+ CD \cos\theta)^2 + BC^2 +(CD\sin\theta)^2 $$

which leads to $\cos\theta =\frac1{\sqrt7}$. Then, the distance from $A$ to the plane $BCD$ is $$d= AB \sin\theta = \sqrt{\frac67}$$

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  • $\begingroup$ Your solution makes sense, but is there a solution without trigonometry? This question appeared on a test that did not require trigonometry, or memorization of any volume formula. Thanks! $\endgroup$
    – user797346
    Apr 2, 2021 at 22:14
  • $\begingroup$ I realized that, I was referring to the other solutions. However, is it possible to remove trigonometry? $\endgroup$
    – user797346
    Apr 2, 2021 at 22:27
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    $\begingroup$ @WWesEEE - you may replace $a = CD\cos\theta$, $b = CD\sin\theta$, where $a$ is the height of D and $b$ is the horizontal distance between C and D. Note $a^2+b^2 = CD^2=28$ $\endgroup$
    – Quanto
    Apr 2, 2021 at 22:36
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You can find the volume of the tetrahedron from the lengths of the edges from this Heron-like formula: $$ {\displaystyle V={\frac {\sqrt {4a^{2}b^{2}c^{2}-a^{2}X^{2}-b^{2}Y^{2}-c^{2}Z^{2}+XYZ}}{12}}} $$ where $a=AB$, $b=AC$, $c=AD$, $x=CD$, $y=BD$, $z=BC$ and $$ {\displaystyle {\begin{aligned}X&=b^{2}+c^{2}-x^{2},\\Y&=a^{2}+c^{2}-y^{2},\\Z&=a^{2}+b^{2}-z^{2}.\end{aligned}}} $$

Just divide the triple of that volume by the area of $BCD$ to find the relative height $AH$.

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I see from comment that you want to avoid trigonometry: so you can follow these steps.

  1. The edges from A form the same angle between them. We can take the z axis from A to be symmetrically placed wrt the three edges. Then they project on the xy plane at $120^\circ$.

  2. Take the three unitary vectors on the xy plane and add the same z component $h$. Take the dot product and determine $h$ for the angle to be $60^\circ$.

  3. Determined $h$ you have the three unit vectors along the edges. Multiply by $1,2,6$ and get the vertices $B,C,D$.

  4. Find the plane through $B,C,D$ and the distance of $A$ from it, or work it out through the determinant of the base (area), and the determinant of the tetrahedron (volume).

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Start with a regular tetrahedron AEFD of edge length 6. Choose B on AE and C on AF at the correct distance. As the ratio AC:AB is 2, BC is perpendicular to AE. D dropped perpendicular to AEF cuts the height over AE of triangle AEF 2:1, the same ratio as C. In other words D lies in the plane through C perpendicular to the height which is parallel to BC, hence DC is perpendicular to BC. The volume of ABCD is 1/18 that of AEFD since that is the ratio of areas ABC and AEF. It is also 1/6 of the product BC x CD x d where d is the distance we are looking for. The volume of regular tetrahedron AEFD is $18 \times \sqrt{2}$, The lengths BC and CD are (here we need a little bit of Pythagoras) $\sqrt{3}$ and $2 \times \sqrt{7}$. Therefore d = $\sqrt{6/7}$.

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