Quadrilaterals
The converse and equal spacing
Prove the converse of the Midpoint Theorem and use it to split a line equally.
The converse: a line through the midpoint of AB, parallel to BC, meets AC at its midpoint (and the piece inside the triangle is BC/2).
A short proof. There is only one line through P parallel to BC. The segment from P to the midpoint M of AC is parallel to BC (Midpoint Theorem), so it is that line, and Q = M.
Equally spaced parallel lines cut equal pieces from any line that crosses them. So on ruled paper, a segment that crosses 4 spaces has its midpoint on the middle line.
Trapezium: the line through the midpoint of one slanting side, parallel to the parallel sides, bisects the other slanting side, and its length is the average of the parallel sides: (AB + CD)/2. Sides 6 and 10 give 8.
Unequal spacing works in proportion. If AP = 1, AQ = √5 and PB = 2, then QC = 2√5: steps of 1 along AB match steps of √5 along AC.
In Grade 10 this becomes the Basic Proportionality Theorem: PQ ‖ BC gives AP/PB = AQ/QC.
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.