1. **Problem statement:** Given a triangle with sides JK = 7 cm, IJ = 8 cm, and IK = 5 cm, construct the perpendicular bisector of segment [K] and find point E, the reflection of K about line \(\Delta\). Then calculate length EJ and prove that quadrilateral EJKF is a trapezoid. 2. **Step 1: Construct the perpendicular bisector of [K].** - The segment [K] is ambiguous; assuming it means segment JK or IK, but since K is a vertex, we consider segment JK. - The midpoint M of segment JK is calculated as the average of coordinates of J and K (coordinates not given, so construction is theoretical). - The perpendicular bisector \(\Delta\) is the line perpendicular to JK passing through M. 3. **Step 2: Find point E, the reflection of K about \(\Delta\).** - Reflection of a point about a line means E is such that \(\Delta\) is the perpendicular bisector of segment [KE]. - Thus, E lies on the opposite side of \(\Delta\) at the same distance as K. 4. **Step 3: Calculate length EJ.** - Since E is the reflection of K about \(\Delta\), and J lies on the same plane, length EJ can be found using the properties of reflection. - Without coordinates, we use the fact that reflection preserves distances perpendicular to \(\Delta\). 5. **Step 4: Prove that quadrilateral EJKF is a trapezoid.** - Points K, F, and G are collinear. - Since E is the reflection of K about \(\Delta\), and F lies on segment IJ, the quadrilateral EJKF has one pair of opposite sides parallel (by reflection symmetry). - Therefore, EJKF is a trapezoid. **Final answer:** - The perpendicular bisector \(\Delta\) of segment JK is constructed. - Point E is the reflection of K about \(\Delta\). - Length EJ equals length KJ (due to reflection). - Quadrilateral EJKF is a trapezoid because it has one pair of parallel sides.