Question

Докажите, что средняя линия описанной равносторонней трапеции равна боковой стороне.

Answer 1

Theorem: The median of an isosceles trapezoid is equal to one of its lateral sides.

To prove this theorem, let's consider an isosceles trapezoid ABCD, where AB and CD are parallel sides, and AD = BC are the non-parallel sides. Let M be the midpoint of the non-parallel sides AD and BC.

To prove that the median of the trapezoid is equal to one of its lateral sides, we need to show that AM = BM.

Proof:

1. Draw diagonals AC and BD of the trapezoid ABCD. These diagonals will intersect at a point O. 2. Since ABCD is an isosceles trapezoid, the diagonals AC and BD are equal in length. Therefore, AO = CO and BO = DO. 3. Since M is the midpoint of AD and BC, we can say that AM = MD and BM = MC. 4. Now, consider the triangles AOM and COM. These triangles share the side AO and the angle at O. Additionally, AM = MD and CO = OB. 5. By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangle AOM is congruent to triangle COM. 6. Therefore, the corresponding sides of these congruent triangles are equal. This means that AM = MC. 7. Similarly, considering the triangles BOM and DOM, we can conclude that BM = MD. 8. Combining the results from steps 6 and 7, we have AM = MC = BM. 9. Hence, the median of the isosceles trapezoid ABCD, represented by the segment connecting the midpoints of the non-parallel sides, is equal to one of its lateral sides.

Therefore, we have proven that the median of an isosceles trapezoid is equal to one of its lateral sides.