Thales' Theorem
Senior 2 · Mathematics
Thales' Theorem
Lesson 1 of 4
20 min
A carpenter building a roof truss nails a horizontal tie-beam between the midpoints of the two sloping rafters. Without measuring, he knows two things: the tie-beam is exactly horizontal (parallel to the base beam) and exactly half as long. That certainty comes from the midpoint theorem, the first step towards Thales' theorem.
Midpoint theorem. In triangle ABC, if M is the midpoint of AB and N is the midpoint of AC, then
(The single tick marks show AM = MB; the double tick marks show AN = NC.)
Extend MN beyond N to a point P so that NP = MN, and join P to C.
A line drawn through the midpoint of one side of a triangle, parallel to a second side, bisects the third side. So if M is the midpoint of AB and a line through M parallel to BC meets AC at N, then N is the midpoint of AC.
1. In triangle ABC, M and N are the midpoints of AB and AC, and BC = 14 cm. Then MN = ½ × 14 = 7 cm.
2. A triangle has sides 10 cm, 12 cm and 14 cm. Joining the midpoints of its three sides forms a smaller triangle. Each side of the small triangle is half of the side it is parallel to: 5, 6 and 7 cm. Its perimeter is 18 cm, exactly half of 36 cm.
3. The roof truss. The base beam of a truss is 8 m long. The tie-beam joining the midpoints of the rafters is 4 m long and parallel to the base.
Tip
Exam technique. Look for tick marks or the words "midpoint" or "bisects". Two midpoints on two sides of one triangle means: parallel to the third side, and half its length. In a proof, write the reason in full: "(midpoint theorem)".
M and N are the midpoints of PQ and PR. MN = 4.5 cm. Find QR.
Solution: MN = ½ QR, so QR = 9 cm.
A triangle has sides 8 cm, 10 cm and 12 cm. Find the perimeter of the triangle formed by joining the midpoints of its sides.
Solution: The sides are halved: 4, 5 and 6 cm. Perimeter = 15 cm.
In triangle XYZ, K is the midpoint of XY. A line through K parallel to YZ meets XZ at L. XZ = 11 cm. Find XL.
Solution: By the converse, L is the midpoint of XZ, so XL = 5.5 cm.
Explain why the midpoints of the four sides of any quadrilateral ABCD form a parallelogram.
Solution: Draw the diagonal AC. In triangle ABC, the segment joining the midpoints of AB and BC is parallel to AC and half its length. In triangle ADC, the segment joining the midpoints of AD and DC is also parallel to AC and half its length. So two opposite sides of the midpoint quadrilateral are equal and parallel, which makes it a parallelogram.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
