Mid-point Theorem & its Converse [Including Intercept Theorem]
ICSE Class 9 · Mathematics · Chapter 11
What this chapter covers
- Mid-points, parallel lines and the tools you need
- The mid-point theorem
- Applications of the mid-point theorem
- The converse of the mid-point theorem
- Applications of the converse
- Intercepts made by parallel lines
- Applications of the equal intercept theorem
- Choosing the right theorem in a problem
Key points
- The mid-point theorem: the segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
- Its proof draws a line through one vertex parallel to the opposite side, proves two triangles congruent (ASA), and uses a parallelogram with one pair of sides equal and parallel.
- The converse: a line through the mid-point of one side, parallel to a second side, bisects the third side. Its proof uses the same construction, a parallelogram and AAS congruence.
- Use the mid-point theorem when two mid-points are known; use the converse when one mid-point and a parallel line are known.
- Joining the mid-points of the sides of a triangle gives four congruent triangles, and the inner triangle has half the perimeter of the original.
- Joining the mid-points of the sides of any quadrilateral gives a parallelogram whose perimeter equals the sum of the diagonals.
- In a trapezium, the segment joining the mid-points of the non-parallel sides is parallel to the parallel sides and equal to half their sum.
- The mid-point of the hypotenuse of a right-angled triangle is equidistant from all three vertices.
- An intercept is the part of a transversal between two of the lines it cuts, measured along the transversal.
- The equal intercept theorem: equal intercepts on one transversal across three or more parallel lines mean equal intercepts on every transversal. It is proved by using the converse of the mid-point theorem twice.
- The equal intercept theorem justifies dividing a line segment into any number of equal parts with a ruler, compass and set square.
Questions answered
State the mid-point theorem.
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
State the converse of the mid-point theorem.
The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
What is an intercept? State the equal intercept theorem.
An intercept is the part of a transversal lying between two of the lines it cuts. If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them also makes equal intercepts.
In the proof of the mid-point theorem, why is the line through C drawn parallel to BA rather than in some other direction?
The parallel line makes ∠ DAE and ∠ FCE alternate angles, which gives the congruence △ ADE cong △ CFE. It also makes DB ∥ CF, so that with DB = CF the figure DBCF is a parallelogram, from which DF ∥ BC and DF = BC follow.
The sides of a triangle are 7 cm, 9 cm and 11 cm. Find the perimeter of the triangle formed by joining the mid-points of its sides.
Each side of the new triangle is half a side of the original, so the perimeter is (1)/(2)(7 + 9 + 11) = 13.5 cm.
In △ XYZ, L and M are the mid-points of XY and XZ. If LM = 6.4 cm and ∠ XLM = 70°, find YZ and ∠ XYZ.
By the mid-point theorem, YZ = 2 × 6.4 = 12.8 cm and LM ∥ YZ, so ∠ XYZ = ∠ XLM = 70° (corresponding angles).
In △ ABC, D is the mid-point of AB, and DE ∥ BC meets AC at E. If AE = 4.5 cm, find AC. Name the theorem used.
By the converse of the mid-point theorem, E is the mid-point of AC, so AC = 2 × 4.5 = 9 cm.
ABCD is a trapezium with AB ∥ DC, AB = 18 cm and DC = 10 cm. E and F are the mid-points of AD and BC. Find EF.
EF = (1)/(2)(AB + DC) = (1)/(2)(18 + 10) = 14 cm.
Past board exam questions
ICSE 2019 sample paper · 4 marks
ABCD is a parallelogram. AB is produced to P, such that AB = BP and PQ is drawn parallel to BC to meet AC produced at Q. Given AB = 8 cm, AD = 5 cm, AC = 10 cm. (i) Prove that point C is mid point of AQ. (ii) Find the perimeter of quadrilateral BCQP.
Show answer
(i) In △APQ, B is the mid-point of AP (AB = BP) and BC ∥ PQ (given). By the converse of the mid-point theorem, the line through the mid-point of one side parallel to another side bisects the third side, so C is the mid-point of AQ.
(ii) BC = AD = 5 cm (opposite sides of parallelogram). By the mid-point theorem, BC = (1)/(2)PQ, so PQ = 10 cm.
CQ = AC = 10 cm (C is the mid-point of AQ); BP = AB = 8 cm.
Perimeter of BCQP = BC + CQ + QP + PB = 5 + 10 + 10 + 8 = 33 cm.