In triangle DFE, since you mentioned that all side lengths are congruent, triangle DFE is an equilateral triangle. This means that each side length is equal, and each angle is 60 degrees.
Given that the lines drawn from each vertex (D, F, E) to the opposite side form right angles at point K, point K represents the foot of the altitude of the triangle. For an equilateral triangle, the lengths of the segments from the centroid (which is also the orthocenter and circumcenter in this case) to the vertices or the sides have specific relationships:
In an equilateral triangle:
The lengths of the segments from the vertices to the opposite sides (the altitudes) are perpendicular to the respective sides and meet at the orthocenter (point K) of the triangle. In an equilateral triangle, the centroid divides each altitude in a ratio of 2:1, where the distance from the vertex to the centroid is double that from the centroid to the side.
Since you provided that KJ (distance from K to side DFE, which is the height from point K to side DE, or EK) is 50 units, we can determine the length of segment EK using the ratio provided by the centroid.
Let:
- Length of segment DK (from D to K) = 2x
- Length of segment EK (from E to K) = x
Based on the centroid property:
- DK = 2 * EK
- KJ = 50 = DR (which stands for the distance from K perpendicular to side DE)
So: DK = KJ + EK = 50 + x = 2x
Thus: 50 + x = 2x => 50 = 2x - x => x = 50
Now substituting x back into the equation for EK: EK = 50 units.
So, the length of segment EK is 50 units.