Question

For the points P(t,n) and Q(n,t), fund (a) the distance formula between P and Q and (b) the coordinates of the midpoint of the segment PQ

Answer 1

The distance between points P(t,n) and Q(n,t) is given by sqrt(2*(n - t)²), and the midpoint is ((t+n)/2, (n+t)/2). The distance remains invariant under rotations due to the preservation of distances and angles in Euclidean space.

Step-by-step explanation:

Distance Formula and Midpoint

To answer the student's question:

• For the points P(t,n) and Q(n,t), the distance formula between P and Q is derived from the Pythagorean theorem and is given by: sqrt((n - t)² + (t - n)²), which simplifies to sqrt(2*(n - t)²).
• The coordinates of the midpoint of segment PQ are found by averaging the coordinates of P and Q, leading to ((t+n)/2, (n+t)/2).
• To show that the distance between points P and Q is invariant under rotations of the coordinate system, you'd use the property that distances and angles are preserved under rotation. The squared distance remains the same before and after the rotation, as it depends only on the difference in coordinates, which doesn't change with rotation.

The solutions to (a) and (b) demonstrate practical examples of the properties of Euclidean geometry.