two edges PQ,RS of a tetrahedron PQRS are perpendicular,show that the distance between the mid-points of PS and QR is equal to the distance between the mid points of PR and QS
To prove that the distance between the midpoints of PS and QR is equal to the distance between the midpoints of PR and QS, we can use vector algebra.
Let's denote the vertices of the tetrahedron as P, Q, R, and S. We also denote the midpoint of PS as M₁, the midpoint of QR as M₂, the midpoint of PR as M₃, and the midpoint of QS as M₄.
First, let's find the coordinates of the midpoints M₁ and M₂. The coordinates of M₁ can be found by averaging the coordinates of P and S:
M₁ = (P + S) / 2
Similarly, the coordinates of M₂ can be found by averaging the coordinates of Q and R:
M₂ = (Q + R) / 2
Next, let's find the vectors PS and QR. We can obtain them by subtracting the coordinates of the vertices:
PS = S - P
QR = R - Q
Since PS and QR are perpendicular, their dot product is zero:
PS · QR = 0
Expanding the dot product using the properties of vector algebra:
(S - P) · (R - Q) = 0
Simplifying the equation:
(S · R) - (S · Q) - (P · R) + (P · Q) = 0
Rearranging the terms:
(S · R) + (P · Q) = (S · Q) + (P · R)
Now let's consider the distances between the midpoints:
Distance between M₁ and M₂:
||M₁ - M₂|| = ||((P + S) / 2) - ((Q + R) / 2)||
Expanding the equation and simplifying:
||(P + S - Q - R) / 2||
Since the distance between M₁ and M₂ is the magnitude of the vector P + S - Q - R, we can rewrite the equation:
||P + S - Q - R|| / 2
Likewise, the distance between M₃ and M₄ can be written as:
||P + R - Q - S|| / 2
Now, let's substitute the values of P + S - Q - R and P + R - Q - S from earlier:
||(S · R) + (P · Q)|| / 2
||-(S · Q) - (P · R)|| / 2
Since (S · R) + (P · Q) = (S · Q) + (P · R), we conclude that:
||M₁ - M₂|| = ||M₃ - M₄||
Therefore, the distance between the midpoints of PS and QR is equal to the distance between the midpoints of PR and QS.