Quadrilaterals · Ages 14–15
The mid-point theorem
The line joining two mid-points is parallel to the third side and half as long.
How to do it
Method
The line joining two mid-points is parallel to the third side and half as long
BC = 12: MN = 12 ÷ 2 = 6
Another way
Similar triangles
AM : AB = 1 : 2, MN = 12 × 12 = 6
In full
The mid-point theorem: in △ABC, with M the mid-point of AB and N the mid-point of AC, MN // BC and MN = BC ÷ 2 (mid-pt. thm.). BC = 12 cm gives MN = 6 cm; and MN // BC gives ∠AMN = ∠ABC (corr. ∠s). The other way round (from the intercept theorem): M the mid-point of AB and MN // BC with N on AC make N the mid-point of AC. Another way: similar triangles. AM : AB = 1 : 2 and ∠A is common, so △AMN ~ △ABC (ratio of 2 sides, inc. ∠) and MN : BC = 1 : 2, the same. The triangle joining the three mid-points has each side half the side facing it, so half the perimeter.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Show the answer and working
Answer: 5 cm
Mid-point theorem: MN = BC ÷ 2, so x = 5.
Example 2
In △ABC, M and N are the mid-points of AB and AC, MN = (x + 4) cm and BC = (4x − 8) cm. x =
Show the answer and working
Answer: 8
BC = 2 × MN (mid-pt. thm.): 4x − 8 = 2(x + 4), x = 8. Check: MN = 12 cm.
Example 3
In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Show the answer and working
Answer: 75°
MN // BC (mid-pt. thm.), so x = ∠ABC = 75° (corr. ∠s, MN // BC).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Answer:
2.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Answer:
3.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
Answer:
4.The quadrilateral ABCD has diagonals AC = 11 cm and BD = 7 cm. P, Q, R and S are the mid-points of AB, BC, CD and DA. The perimeter of PQRS =
Answer:
5.In the trapezium ABCD, AB // DC, and M and N are the mid-points of AD and BC. x =
Answer:
All the answers
- 1. 6 cm
- 2. 8 cm
- 3. 80°
- 4. 18 cm
- 5. 6 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake