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Maths > Quadrilaterals > Midpoint Theorem and Equal Intercept Theorem
Quadrilaterals

Midpoint Theorem and Equal Intercept Theorem

Do you know what midpoint theorem is?  What is the purpose of using the midpoint theorem? Yes, it is used in solving geometry problems related to triangles. So let us study the midpoint theorem and equal intercept theorem in detail.

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Midpoint Theorem

Intercept

The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side. Let us observe the figure given above. We can see the ΔABC,

Point D and E are the midpoints of side AB and side AC respectively. Also, the segment DE connects the two sides at the midpoints, then DE || BC and DE is half the length of side BC.

If we know the length of BC then it is very convenient to find the length of DE as DE is half of BC. It also allows us to find the length of sides AE, EC, BD, and DA. Since DE is parallel to BC we know that the distance between these two line segment is equal.

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Also, ∠ADE = ∠ABC

So, DE || BC

Proof of the Theorem

Intercept

Given: In triangle ABC, D and E are midpoints of AB and AC respectively.

To Prove:

  1. DE || BC
  2. DE = 1/2 BC

Construction: Draw CR || BA to meet DE produced at R. (Refer the above figure)

∠EAD = ∠ECR. (Pair of alternate angles) ———- (1)
AE = EC. (∵ E is the mid-point of side AC) ———- (2)
∠AEP = ∠CQR (Vertically opposite angles) ———- (3)
Thus, ΔADE ≅ ΔCRE (ASA Congruence rule)

DE = 1/2 DR ———- (4)
But, AD= BD. (∵ D is the mid-point of the side AB)
Also. BD || CR. (by construction)

In quadrilateral BCRD, BD = CR and BD || CR
Therefore, quadrilateral BCRD is a parallelogram.
BC || DR or, BC || DE
Also, DR = BC (∵ BCRD is a parallelogram)
⇒ 1/2 DR = 1/2 BC

The Converse of MidPoint Theorem

The line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side.

Equal Intercept Theorem

The theorem states if a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts. It means that given any three mutually perpendicular lines, a line passing through them forms intercepts in the corresponding ratio of the distances between the lines.

For example, Suppose there are three lines, l, m and n. Keep the distance between lm twice than the distance between mn. So any line passing through them, the intercept made by l-m on the line is twice the intercept made by m-n.

Solved Examples

Q1. In the adjoining figure, all measurements are indicated in centimetre. Find the length of AO if AX =9.5 cm.

Intercept

  1. 6.75 cm
  2. 9.25 cm
  3. 4.25 cm
  4. 4.75 cm

Solution: D. Given in the  figure AD = DB = 4
AE =EC = 6
Then D and E are the midpoints of sides AB and AC respectively.
Thus, DE bisects AX at point O
∴ AO = 1/2 AX = 1/2 × 9.5 = 4.75
Hence the length of AO = 4.75

Q2. In the given Δ ABC, Z is the mid-point of the median AD. If the area Δ ABC  is 18 m²,  find the area of ΔBZC.

  1. 7  m²
  2. 9  m²
  3. 6  m²
  4. 5  m²

Solution: B. In Δ ABC, Z is the mid-point of the median AD
Join Z to B and C
∴ it will divide Δ ABC into two triangles of equal area.
∴ Area of  ΔBZD = Area of  ΔBZA
A(ΔBZD) = 1/2 A(ΔABD)
Also, A((ΔZDC)  = A(ΔAZC)
= A((ΔZDC)  = 1/2 A(ΔAZC)
A(ΔBZD) + A((ΔZDC) =1/2  A((ΔABD) + 1/2 A((ΔADC)
A(ΔBZC) = 1/2 A(ΔABC) = 1/2 × 18 = 9m²

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