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.
Step-by-step solution
To find: Show that the crease passes through the midpoints of AB and AC
Idea: A fold that puts one point on another creates the perpendicular bisector of the segment joining them. The right angle at B then makes the crease parallel to BC.
- When A is folded onto B, every point of the crease is the same distance from A as from B, and the crease meets AB at right angles at its midpoint M. So M (the midpoint of AB) is found, and the crease ⟂ AB.1 mark
- ∠ABC = 90°, so BC ⟂ AB. Two lines perpendicular to the same line AB are parallel: crease ‖ BC.1 mark
- In ∆ABC the crease passes through the midpoint M of AB and is parallel to BC, so by the converse of the Midpoint Theorem it bisects AC. Where the crease meets AC is the midpoint N of AC.1 mark
Answer to write in the exam
Folding A onto B ⇒ crease is the perpendicular bisector of AB ⇒ passes through midpoint M of AB, crease ⟂ AB
∠B = 90° ⇒ BC ⟂ AB ⇒ crease ‖ BC
In ∆ABC: line through M ‖ BC bisects AC (converse of Midpoint Theorem)
∴ The crease meets AC at its midpoint N.
Common mistakes that cost marks
- Assuming the crease passes through the midpoint of AC without the parallel-line argument.
- Forgetting why the crease is parallel to BC: it uses the right angle at B.
- Thinking the crease bisects BC. It is parallel to BC; it bisects AB and AC.
How this can come in the exam
A tailor has a piece of cloth shaped like a right triangle ABC with ∠B = 90°, AB = 60 cm and BC = 80 cm. She folds it so that corner A lies on corner B and presses a crease.
(i) How far from B does the crease cross AB? (ii) Is the crease parallel to BC? Why? (iii) How long is the crease inside the cloth? (iv) Find AC and the distance from A to where the crease meets AC.
Show answer
(i) The crease is the perpendicular bisector of AB, so it crosses AB at 30 cm from B (1 mark). (ii) Yes: crease ⟂ AB and BC ⟂ AB (1 mark). (iii) It joins the midpoints of AB and AC, so its length is ½BC = 40 cm (1 mark). (iv) AC = √(60² + 80²) = 100 cm; the crease meets AC at its midpoint, 50 cm from A (1 mark).Try one yourself
In right ∆PQR (∠Q = 90°), the paper is folded so that R falls on Q. Which two midpoints does the crease pass through?
Show answer
The midpoints of QR and PR (the crease is ⟂ QR, hence parallel to PQ, and passes through the midpoint of QR).
More questions like this
- 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.
- The diagonals AC and BD of a parallelogram ABCD intersect at O. A line through O meets AB and CD at points P and Q respectively. Show that O is the midpoint of PQ. (Multiple proofs are possible. Which is the simplest?)
- ABCD is a trapezium with parallel sides AD = 3 cm and BC = 5 cm. E and F are the midpoints of the non-parallel sides. Find the ratio of the areas of the 4-gons AEFD and EBCF.
- Consider 4 points A, B, C, D in the plane with no three collinear. Answer the following questions. Some answers may require you to consider different cases, depending on how the points are positioned in the plane.
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