The Mid Point Theorem
Mid Point Theorem
Geometry is an important part of mathematics that deals with different shapes and figures. Triangles are an important part of geometry and the mid-point theorem points towards mid points of the triangle.What is Mid-Point Theorem?
This theorem states that" The line segment joining mid-points of two sides of a triangle is parallel to the third side of the triangle and is half of it"
Proof of Mid-Point Theorem
A triangle ABC in which D is the mid-point of AB and E is the mid-point of AC.
To Prove: DE ∥ BC and DE = 1/2(BC)
Construction
Extend the line segment joining points D and E to F such that DE = EF and join CF.
Proof
In ∆AED and ∆CEF
DE = EF (construction)
∠1 = ∠2 (vertically opposite angles)
AE = CE (E is the mid-point)
△AED ≅ △CEF by SAS criteria
Therefore,
∠3 =∠4 (c.p.c.t)
But these are alternate interior angles.
So, AB ∥ CF
AD = CF(c.p.c.t)
But AD = DB (D is the mid-point)
Therefore, BD = CF
In BCFD
BD∥ CF (as AB ∥ CF)
BD = CF
BCFD is a parallelogram as one pair of opposite sides is parallel and equal.
Therefore,
DF∥ BC (opposite sides of parallelogram)
DF = BC (opposite sides of parallelogram)
As DF∥ BC, DE∥ BC and DF = BC
But DE = EF
So, DF = 2(DE)
2(DE) = BC
DE = 1/2(BC)
Hence, proved that the line joining mid-points of two sides of the triangle is parallel to the third side and is half of it.
What is the Converse of Mid-Point Theorem?
The line drawn through the mid-point of one side of a triangle parallel to the base of a triangle bisects the third side of the triangle.
Proof of the Theorem
In triangle PQR, S is the mid-point of PQ and ST ∥ QR
To Prove: T is the mid-point of PR.
Construction
Draw a line through R parallel to PQ and extend ST to U.
Proof
ST∥ QR(given)
So, SU∥ QR
PQ∥ RU (construction)
Therefore, SURQ is a parallelogram.
SQ = RU(Opposite sides of parallelogram)
But SQ = PS (S is the mid-point of PQ)
Therefore, RU = PS
In △PST and △RUT
∠1 =∠2(vertically opposite angles)
∠3 =∠4(alternate angles)
PS = RU(proved above)
△PST ≅ △RUT by AAS criteria
Therefore, PT = RT
T is the mid-point of PR.
Sample Problems on Mid Point Theorem
Problem 1: l, m, and n are three parallel lines. p and q are two transversals intersecting parallel lines at A, B, C, D, E, and F as shown in the figure. If AB:BC = 1:1, find the ratio of DE : EF.
Given: AB:BC=1:1
To find: DE:EF
Construction: Join AF such that it intersects line m at G.
In △ACF
AB = BC(1:1 ratio)
BG∥ CF(as m∥n)
Therefore, by converse of mid-point theorem G is the midpoint of AF(AG = GF)
Now, in △AFD
AG = GF(proved above)
GE∥ AD(as l∥m)
Therefore, by converse of mid-point theorem E is the mid-point of DF(FE = DE)
So, DE:EF = 1:1(as they are equal)
Problem 2: In the figure given below L, M and N are mid-points of side PQ, QR, and PR respectively of triangle PQR.
If PQ = 8cm, QR = 9cm and PR = 6cm. Find the perimeter of the triangle formed by joining L, M, and N.
Solution: As L and N are mid-points
By mid-point theorem
LN ∥ QR and LN = 1/2 * (QR)
LN = 1/2 × 9 = 4.5cm
Similarly, LM = 1/2 * (PR) = 1/2×(6) = 3cm
Similarly, MN = 1/2 * (PQ) = 1/2 × (8) = 4cm
Therefore, the perimeter of △LMN is LM + MN + LN
= 3 + 4 + 4.5
= 11.5cm
Perimeter is 11.5 cm
Result: Perimeter of the triangle formed by joining mid-points of the side of the triangle is half of the triangle.
FAQs on The Mid Point Theorem
| 1. What exactly is the midpoint theorem and how does it work in triangles? |  |
| 2. How do I identify which sides are parallel when using the midpoint theorem? | |
| 3. Why does the midpoint theorem create a segment that's exactly half the length? | |
| 4. Can the midpoint theorem be applied to quadrilaterals or just triangles? | |
| 5. What's the difference between the midpoint theorem and the converse of the midpoint theorem? | |
