along with the same idea used in the second proof – Discussion Forum

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Virat

Asked: October 3, 2026In: Class 9 Maths

A more general midpoint theorem and its converse. In a quadrilateral ABCD, suppose AB ∥ DC. Recall that such ABCD is called a trapezium. Let E be the midpoint of AD. A line drawn through E intersects side BC at F. (i) If EF ∥ AB, then show that F is the midpoint of BC. Conclude that EF = (AB + CD)/2. (ii) If F is the midpoint of BC, then show that EF ∥ AB. There are at least two ways to solve (ii). Do it directly by using the midpoint M of BD and showing that EM and FM are the same lines. Or use (i) along with the same idea used in the second proof of the converse of the Midpoint Theorem. Which way do you think is simpler?

(i) Diagonal BD meets EF at M. By Theorem 7, M and F bisect BD and BC. Adding segment lengths gives EF = (AB + CD)/2. (ii) Both methods work; using parallel line uniqueness is simpler. Class 9 Ganita Manjari Part ...