Mark the midpoint of the line drawn on the paper (see the figure), given that the horizontal lines are equally spaced. Justify your answer.
Step-by-step solution
Idea: Equally spaced parallel lines cut any slanted segment into equal parts. Build a triangle with the segment as one side and a ruled line as another side, then use the converse of the Midpoint Theorem.
- Call the ends X (lower) and Y (upper). Counting the ruled lines, Y is 4 spaces above X, so the middle line is 2 spaces above X. Let it cross XY at K.1 mark
- Draw XZ perpendicular to the lines, with Z on Y’s line. The lines are equally spaced, so the middle line crosses XZ at its midpoint W (XW = WZ = 2 spaces).1 mark
- In ∆XZY, the middle line passes through W, the midpoint of XZ, and is parallel to ZY (both are ruled lines). By the converse of the Midpoint Theorem it bisects XY. So K is the midpoint of XY.1 mark
Answer to write in the exam
XY runs from a ruled line to the line 4 spaces above; middle line is 2 spaces from each
Draw XZ ⟂ lines, Z on Y’s line; middle line meets XZ at W with XW = WZ (equal spacing)
In ∆XZY: W midpoint of XZ, middle line ‖ ZY ⇒ it bisects XY at K (converse of Midpoint Theorem)
∴ K, where XY crosses the middle line, is the midpoint of XY.
Common mistakes that cost marks
- Marking the midpoint halfway along the paper instead of halfway along the segment.
- Counting lines instead of spaces: the ends are 4 spaces apart, so the middle is 2 spaces from each end.
- Justifying only with a ruler measurement; the question asks for a reason (equal spacing and the converse of the Midpoint Theorem).
How this can come in the exam
A ladder leans against a wall and crosses the horizontal rails of a fence that are 30 cm apart. Its foot is on the ground rail and it crosses the 6th rail above the ground at point T. Where is the midpoint of the part of the ladder from the ground to T?
Show answer
The rails are equally spaced and parallel, so they cut the ladder into equal parts (1 mark). The midpoint is where it crosses the 3rd rail above the ground, 90 cm up (1 mark).Try one yourself
On ruled paper, a segment runs from one line to the line 6 spaces above it. Which lines divide it into three equal parts?
Show answer
The lines 2 and 4 spaces above the lower end (equally spaced parallel lines make equal intercepts).
More questions like this
- You know that the sum of angles of a quadrilateral is 360°, even for a non-convex quadrilateral. (Recall the proof.) Now consider a self-intersecting quadrilateral ABCD, where AB and CD intersect at point E. Show that ∠A + ∠B + ∠C + ∠D < 360°. Can you construct ABCD such that ∠A + ∠B + ∠C + ∠D = 2°?
- In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the figure). Show that APCQ is a parallelogram.
- A right-triangle shaped cutout of a paper is folded such that point A touches point B (see the figure). Show that the crease line can be used to find the midpoint of not only AB but also that of AC.
- You saw how to use the Midpoint Theorem to divide a given triangle into 4 congruent triangles. Can we divide a triangle into 3 congruent triangles? This exercise shows us how to do that and more, provided we are allowed to cut and reassemble.
- 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.
All Quadrilaterals and parallelograms questions · All maths questions