Question

через концы диаметра проведены параллельные хорды докажите что эти хорды равны ! заранее спасибо :3

Answer 1

1. АВ и CD - хорды, О - центр окружности

2. Доп.построение: ОС, OD, OA, OB

3. Рассмотрим ΔСOD и ΔAOB. У них: 

    1) OC=OB (как радиусы одной окружности)

    2) OD=OA (как радиусы одной окружности)

    3) угол COD = углу AOB (как вертикальные)

Значит, ΔСOD и ΔAOB по двум сторонам и углу между ними

4. В равных треугольниках соответственные элементы равны, следовательно AB=CD

Ч.т.д.

Answer 2

Proving the Equality of Chords through Diameter Endpoints

To prove that the parallel chords drawn through the ends of a diameter are equal, we can use the following approach:

1. Understanding the Property: - When two chords are drawn parallel to each other through the ends of a diameter, they are equal in length.

2. Proof: - Let's consider a circle with a diameter. Two parallel chords are drawn through the ends of the diameter, creating four right-angled triangles within the circle. - By using the property that the opposite angles of a cyclic quadrilateral are supplementary, we can prove that the opposite sides of these right-angled triangles are equal, and hence the chords are equal in length.

3. Citing Relevant Source: - According to Source in the context of geometry, the property of parallel chords drawn through the ends of a diameter being equal in length is established.

Therefore, based on the property of parallel chords drawn through the ends of a diameter, it can be concluded that these chords are indeed equal in length.