Angle ABC is a right triangle with hypotenuse AB

Question

Given: Angle ABC is a right triangle with hypotenuse AB D,E,and F are midpts of the sides of the triangle Prove: a. CDEF is a rectangle b. EC is half of AB

Answer 1

a. ADE is similar to ACB because AD/AC=AE/AB=1/2 by definition of midpoints, and angle A is the common included angle; therefore DE is parallel to CB and ADE is a right angle.

EBF is similar to ABC because EB/AB=BF/BC=1/2 by definition of midpoints, and angle B is the common included angle; therefore EF is parallel to AC and BFE is a right angle. DEF, EFB are right angles (alternate angles between parallel lines. So CDEF is a rectangle.

b. EC is a diagonal of CDEF. In the triangles CEF and BEF, EF is a common side. CF=FB because F is the midpoint of CB. Angles EFC and EFB are right angles, so the triangles are congruent (2 sides and the included angle) and CE  (EC) must therefore be equal to EB=(1/2)AB.