if in a tetrahedron the sums of the squares of opposite sides are equal,show that its pairs of opposite sides are right angles
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To prove that pairs of opposite sides in a tetrahedron are right angles when the sums of the squares of opposite sides are equal, we can use vector algebra.
Let's consider a tetrahedron with vertices A, B, C, and D. We denote the positions of these vertices as vector rA, rB, rC, and rD, respectively.
We can define the vectors connecting the vertices of the tetrahedron as follows:
AB = rB - rA
AC = rC - rA
AD = rD - rA
BC = rC - rB
BD = rD - rB
CD = rD - rC
Now, let's express the condition "the sums of the squares of opposite sides are equal" mathematically. We have:
|AB|^2 + |CD|^2 = |AC|^2 + |BD|^2
Using the definition of vector norm (magnitude), we can rewrite this equation as:
(AB · AB) + (CD · CD) = (AC · AC) + (BD · BD)
where · represents the dot product of two vectors.
Now, let's prove that if the above equation holds, then pairs of opposite sides are right angles.
From the definition of the dot product, we know that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them:
AB · AB = |AB| * |AB| * cos(∠AB)
CD · CD = |CD| * |CD| * cos(∠CD)
AC · AC = |AC| * |AC| * cos(∠AC)
BD · BD = |BD| * |BD| * cos(∠BD)
Using these definitions, we can rewrite the equation as:
|AB|^2 * cos(∠AB) + |CD|^2 * cos(∠CD) = |AC|^2 * cos(∠AC) + |BD|^2 * cos(∠BD)
Since the equation holds for any tetrahedron, we can focus on specific pairs of vectors that correspond to opposite sides.
Let's consider the vectors AB and CD. If the equation holds, we have:
|AB|^2 * cos(∠AB) + |CD|^2 * cos(∠CD) = |CD|^2 * cos(∠CD) + |AB|^2 * cos(∠AB)
Simplifying this equation, we get:
|AB|^2 * cos(∠AB) = |CD|^2 * cos(∠CD)
Since the magnitudes |AB| and |CD| are non-zero, this equation implies that the angles must be equal:
cos(∠AB) = cos(∠CD)
For this equality to hold, the angles ∠AB and ∠CD must be either both acute or both obtuse. In other words, they must be equal or supplementary.
If ∠AB = ∠CD, then AB and CD are parallel and opposite, and they form a right angle with the plane containing the other two edges of the tetrahedron.
If ∠AB + ∠CD = 180 degrees, then AB and CD intersect, forming a dihedral angle of 180 degrees. This is also a right angle.
Since any pair of opposite sides in a tetrahedron falls into one of these two categories, we conclude that if the sums of the squares of opposite sides are equal, then the pairs of opposite sides are right angles.