The midsegment of a triangle: parallel to the third side and half its length
Mathematics Prep 2 Term 1, Unit 3, Lessons 3–4
GoalUse the midsegment: it is parallel to the third side and half as long, to find lengths, perimeters and sides of shapes made from midpoints.
New words
midsegment · the segment joining the midpoints of two sides of a trianglethird side · the side the midsegment does not touch; it is the one the midsegment is parallel toperimeter · the sum of the lengths of all the sidesparallelogram · a four-sided shape with both pairs of opposite sides parallel
Explanation
The midsegment DE joins the midpoints D of AB and E of AC. By the last lesson, the line through D parallel to BC passes through the midpoint of AC, and that is E. So DE ∥ BC. Moreover DE = ½ BC. A triangle has three midsegments; they cut it into four small triangles that are all the same size. The half rule uses the side the midsegment is parallel to: if BC = 12 cm, then DE = 6 cm. It is not half of AB or AC. Conversely, if DE is a midsegment and DE = 5 cm, the side parallel to it is BC = 10 cm. Both ends of the segment must be midpoints. A segment drawn parallel to BC from a point that is not a midpoint is not half of BC.
Worked examples
The sides of Δ ABC are 10 cm, 14 cm and 18 cm. D, E and F are the midpoints of the sides. The midsegments are half of the sides they are parallel to: 5 cm, 7 cm and 9 cm. The perimeter of Δ DEF is 5 + 7 + 9 = 21 cm, which is half of the perimeter of Δ ABC (42 cm).
A right-angled triangle has legs 6 cm and 8 cm. The hypotenuse is √(36 + 64) = 10 cm. The midsegment parallel to the hypotenuse is 5 cm, the one parallel to the 8 cm leg is 4 cm, and the one parallel to the 6 cm leg is 3 cm. These three halves, 3, 4 and 5, make a right-angled triangle again.
In the quadrilateral ABCD the diagonals are AC = 10 cm and BD = 14 cm. P, Q, R and S are the midpoints of AB, BC, CD and DA. In Δ ABC, PQ is a midsegment, so PQ = 5 and PQ ∥ AC. In Δ ADC, SR is a midsegment, so SR = 5 and SR ∥ AC. So PQ and SR are equal and parallel. In the same way QR = PS = 7 cm (half of BD). Perimeter of PQRS = 5 + 7 + 5 + 7 = 24 cm. PQRS is a parallelogram.
Class activity
Fold a triangle. Cut a paper triangle. Fold one corner onto the opposite side so that the crease runs through the midpoints of two sides. Measure the crease and the third side. How do they compare? Do the same for a second triangle.
Practice
1
In Δ ABC, D and E are the midpoints of AB and AC and BC = 18 cm. Find DE.
DE = ½ × 18 = 9 cm.
2
The midsegment DE is 7.5 cm and is parallel to BC. Find BC.
BC = 2 × 7.5 = 15 cm.
3
A triangle has sides 12, 16 and 20 cm. Find the perimeter of the triangle formed by the three midpoints.
The sides are 6, 8 and 10 cm, so the perimeter is 24 cm (half of 48).
4
In Δ ABC, D and E are midpoints of AB and AC, DE = 2x + 3 and BC = 5x − 4. Find x and BC.
BC = 2 DE, so 5x − 4 = 4x + 6 and x = 10. BC = 46.
5
In a quadrilateral the diagonals are 8 cm and 6 cm. Find the perimeter of the quadrilateral made by the midpoints of the sides.
Its sides are 4, 3, 4 and 3 cm, so the perimeter is 14 cm.
