Question

Площина а перетинає ребра AD, BD, BC і АС трикутної піраміди DABC у точках E, F, P i Q відповідно, EP=FQ. Ця площина паралельна прямим AB і CD. Доведіть, що чотирикутник EFPQ - прямокутник

Answer 1

Given Information

The problem provides the following information: - There is a plane that intersects the edges AD, BD, BC, and AC of the triangular pyramid DABC at points E, F, P, and Q respectively. - EP = FQ. - This plane is parallel to the lines AB and CD. We are asked to prove that the quadrilateral EFPQ is a rectangle.

Proof

To prove that the quadrilateral EFPQ is a rectangle, we can use the given information and the properties of parallel lines and planes.

1. Opposite Sides of a Rectangle - In a rectangle, opposite sides are equal in length and parallel to each other.

2. Parallel Planes and Intersecting Lines - When a plane intersects two parallel lines, the corresponding segments on the lines are proportional.

3. Using the Given Information - Since the plane is parallel to the lines AB and CD, the segments EP and FQ are proportional to the corresponding segments on AB and CD.

4. Applying the Properties - As EP = FQ and the plane is parallel to AB and CD, we can conclude that the segments EP and FQ are equal in length.

5. Conclusion - Therefore, the quadrilateral EFPQ has opposite sides that are equal in length, making it a parallelogram. Since it also has a right angle due to the intersection with the base of the pyramid, it meets the definition of a rectangle.

Hence, we have proved that the quadrilateral EFPQ is a rectangle.

This proof demonstrates that the given conditions lead to the conclusion that the quadrilateral EFPQ is a rectangle.