Draw a triangle and mark the mid-points E and F of two sides of the triangle. Join the points E and F in following fig.
Measure EF and BC. Measure ∠ AEF and ∠ ABC. EF = `1/2` BC and ∠ AEF = ∠ ABC
so, EF || BC
Theorem : The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Observe following Fig. in which E and F are mid-points of AB and AC respectively and CD || BA.
∆ AEF ≅ ∆ CDF (ASA Rule)
So, EF = DF and BE = AE = DC
Therefore, BCDE is a parallelogram.
This gives EF || BC.
In this case, also note that EF = `1/2` ED = `1/2` BC.
Theorem : The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.
In following fig.
Observe that E is the mid-point of AB, line l is passsing through E and is parallel to BC and CM || BA.
Prove that AF = CF by using the congruence of ∆ AEF and ∆ CDF.
- Theorem of midpoints of two sides of a triangle
- Converse of midpoint theroem
Shaalaa.com | Mid-point theorem
In the given figure, points X, Y, Z are the midpoints of side AB, side BC and side AC of Δ ABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
In the given figure, `square` PQRS and `square` MNRL are rectangles. If point M is the midpoint of side PR then prove that,
(i) SL = LR, (ii) LN = `1/2`SQ
In the given figure, Δ ABC is an equilateral traingle. Points F,D and E are midpoints of side AB, side BC, side AC respectively. Show that Δ FED is an equilateral traingle.
In the given figure, seg PD is a median of Δ PQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that `(PM)/(PR) = 1/3`
[Hint: DN || QM]
In the given Figure, `square` ABCD is a trapezium. AB || DC .Points P and Q are midpoints of seg AD and seg BC respectively.
Then prove that, PQ || AB and PQ = `1/2 (AB + DC)`.