Question

Select the correct answer from each drop-down menu. Given: Prove that three lines AD, CF, and BE are intersecting each other at the midpoint O. Complete the proof. It is given that ________ and ________. By the ________, ________. Therefore, ________. By the ________, ________, and by the ________, ________. After application of the ________, ________.

Answer 1

To prove that three lines AD, CF, and BE intersect at the midpoint O, we assume they are medians of a triangle. By the Median Theorem and Intersection Theorem, these lines are concurrent at the centroid, which is also the midpoint O.

Step-by-step explanation:

To prove that three lines AD, CF, and BE intersect at the midpoint O, we need to rely on certain geometric properties and theorems. Let's assume it is given that lines AD, CF, and BE are medians of a triangle, meaning they connect vertices of the triangle to the midpoints of the opposite sides. By the definition of medians of a triangle, we know that medians intersect at the centroid of the triangle, which is also the midpoint O. Therefore, AD, CF, and BE will intersect at O.

By the Median Theorem, the centroid divides each median into two segments, with the segment closest to the vertex being twice as long as the segment closest to the midpoint of the triangle's side. And by the Intersection Theorem, if three or more lines intersect at a single point, they are said to be concurrent, which further supports our proof. After applying geometric properties and theorems, we can conclude that AD, CF, and BE do indeed intersect at midpoint O.