Mid-point and Intercept Theorem

Take any triangle ABC. Mark P, the mid-point of side AB, and Q, the mid-point of side AC. Join P to Q. The mid-point theorem makes two claims about this little segment: PQ is parallel to the third side BC, and its length is exactly half of BC. Both facts hold for every triangle, no matter its shape. This single result is one of the most useful tools in Class 9 geometry, because it instantly links a mid-point line to a side that may be far away.

The converse turns the theorem around. Start again from the mid-point P of AB, but this time draw a line through P that is parallel to BC. Where does it meet AC? The converse guarantees it meets AC exactly at its mid-point. So a line through one mid-point, parallel to a second side, automatically bisects the third side. This is handy when you know a line is parallel and want to prove a point is a mid-point without measuring.

Now move from triangles to parallel lines. Suppose three or more parallel lines are cut by a transversal, and the pieces it cuts off (the intercepts) are all equal. The equal intercept theorem says that any other transversal crossing the same parallel lines is also cut into equal pieces. It follows from the mid-point theorem and is the idea behind dividing a line segment into several equal parts with ruler and compass.

These are not just facts to memorise; in Selina ICSE Class 9 you are expected to prove them and then use them. A favourite application: join the mid-points of the four sides of any quadrilateral and the new figure is always a parallelogram. The proof uses the mid-point theorem twice on the two triangles formed by a diagonal. Learning to chain the theorem like this is the real skill the chapter trains.