Mid-point Theorem and Intercept Theorem

Mid-point Theorem Let’s try the following activity. First draw a triangle. Mark the mid-points of any two sides of the triangle. What do you think about the properties between the line segment formed and the third side of the triangle? Then join the two points with a line segment.

Mid-point Theorem The mid-point theorem below states the properties of the line joining the mid-points of two sides of a triangle. A B C M N The line segment joining the mid-points of two sides of a triangle is parallel to the third side, and is half the length of the third side. i.e. In △ABC, if AM = MB and AN = NC, then (i) MN // BC, BC. MN  = 2 1 (ii) [Abbreviation: mid-pt. theorem]

In the figure, D and E are mid-points of AB and AC respectively. 73 ∵ AD = DB and AE = EC y D E x cm 2 1  = BC DE ∴ (mid-pt. theorem) z 55 B C 8 cm 8 2 1  = x 4 = also DE // BC (mid-pt. theorem) ∴ AED = ACB (corr. s, DE // BC) y = 55 also ABC = ADE (corr. s, BC // DE) z = 73

Follow-up question 2 cm A B C E D 6 cm The figure shows △ABC. D and E are mid-points of AB and AC respectively. Find the area of △ABC. Solution ∵ AD = DB and AE = EC (mid-pt. theorem) 2 1  = BC DE ∴ BC = 2  DE = 2  2 cm = 4 cm

Follow-up question (cont’d) 2 cm A B C E D 6 cm The figure shows △ABC. D and E are mid-points of AB and AC respectively. Find the area of △ABC. Solution ∵ DE // BC (mid-pt. theorem) ∴ ABC = ADE (corr. s, BC // DE) = 90

Follow-up question (cont’d) 2 cm A B C E D 6 cm The figure shows △ABC. D and E are mid-points of AB and AC respectively. Find the area of △ABC. Solution 2 1  = BC  AB ∴ Area of △ABC 2 1  = 4  6 cm2 = 12 cm2

Intercept Theorem M In the figure, the transversal MN cuts three straight lines L1, L2 and L3 at points A, B and C respectively. transversal L1 A intercept B L2 AB is called the intercept made by L1 and L2 on MN, while BC is called the intercept made by L2 and L3 on MN. C L3 N Note that a transversal is a straight line while an intercept refers to the length of a line segment.

If the intercepts made by three or more parallel If the intercepts are made by parallel lines, we have the intercept theorem. If the intercepts made by three or more parallel lines on the same transversal are equal, then the intercepts made by these parallel lines on any other transversals are also equal. i.e. If L1 // L2 // L3 then DE = EF. [Abbreviation: intercept theorem] A B C D E F L1 L2 L3 and AB = BC,

In the figure, ACE and BDF are straight lines. 8 cm ∵ AB // CD // EF and AC = CE ∴ BD = DF (intercept theorem) 2 1  = BF BD i.e. cm 8 2 1  = cm 4 = ∴ DF = 4 cm

Follow-up question In the figure, MPS and NQT are straight lines. Find the lengths of PS and MS. M N 5 cm P Q Solution S T ∵ MN // PQ // ST and NQ = QT ∴ PS = MP (intercept theorem) = 5 cm MS = MP + PS = 5 cm + 5 cm = 10 cm

The following shows a special case of the intercept theorem. If we consider △ABC, we can deduce the intercept theorem for triangles. The following shows a special case of the intercept theorem. F C A D E B

[Abbreviation: intercept theorem] Intercept theorem for triangles If a line passes through the mid-point of one side of a triangle and is parallel to another side, it bisects the third side of the triangle. A E F i.e. In △ABC, if AE = EB , and EF // BC, B C then AF = FC. [Abbreviation: intercept theorem] Note: The intercept theorem for triangles is the converse of the mid-point theorem.

∴ AF = FC (intercept theorem) 10 cm In the figure, AEB and AFC are straight lines. E F ∵ EF // BC and AE = EB ∴ AF = FC (intercept theorem) B C 2 1  = AC AF cm 10 2 1  = cm 5 = and FC = 5 cm

Follow-up question In the figure, ADFB and AEGC are straight lines. 3 cm D E Solution x cm Consider △AFG. ∵ AD = DF and DE // FG ∴ EG = AE (intercept theorem) x = 3 ∵ DF = FB and DE // FG // BC ∴ GC = EG (intercept theorem) y = 3 F G y cm B C