Question

Please answer this question​

Answer 1

MN = 1/2(AB+CD)

MN = 1/2(11+8)

=. 1/2×19

=. 9.5

Answer 2

The length of MN is 9.5 cm.

Given:

         ⇒  Length of AB = 11 cm

         ⇒  Length of DC = 8 cm

         ⇒  M, N are midpoints of AD and BC respectively.

         ⇒  AB ║ DC

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We have to recall the concept given below.

Three or more parallel lines cut two or more lines passing through them proportionally.

As M and N are midpoints of AD and BC respectively, AM : MD = BN : NC = 1 : 1. This means AD and BC are cut by the lines AB, MN and DC proportionally.

So it can be said that 'MN ║ AB ║ CD".

AD : DM = BC : CN = 2 : 1.

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Construction:

         ⇒  Draw an altitude of the trapezium from D to point E at AB.

         ⇒  Draw other altitude of the trapezium from C to point F at AB.

         ⇒  Mark the point of intersection of DE and MN as P.

         ⇒  Mark the point of intersection of CF and MN as Q.

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As DE and FC are perpendicular to AB,  EF = PQ = DC = 8 cm.

Let  AE = x

∴ BF = AB - (AE + EF) = 11 - (x + 8) = 11 - x - 8 = 3 - x.  

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Consider triangles AED and MPD.

    →  ∠A = ∠M (corresponding because AB ║ MN)

    →  ∠E = ∠P (corresponding because AB ║ MN)

    →  ∠D = ∠D (common)

∴ ΔAED ~ ΔMPD

∴ AD : DM = AE : MP = x : MP = 2 : 1

∴ MP = x/2

Consider triangles BFC and NQC.

    →  ∠B = ∠N (corresponding because AB ║ MN)

    →  ∠F = ∠Q (corresponding because AB ║ MN)

    →  ∠C = ∠C (common)

∴ ΔBFC ~ ΔNQC

∴ BC : CN = BF : QN = (3 - x)/ QN = 2 : 1

∴ QN = (3 - x) / 2

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MN = MP + PQ + QN

MN = x/2 + 8 + (3 - x) / 2

MN = 8 + (x + 3 - x)/2

MN = 8 + 3/2

MN = 8 + 1.5

MN = 9.5 cm

$\displaystyle MN = MP + PQ + QN \\ \\ \\ MN = \frac{x}{2} + 8 + \frac{3-x}{2} \\ \\ \\ MN = 8 + \frac{x+3-x}{2} \\ \\ \\ MN = 8 + \frac{3}{2} \\ \\ \\ MN = 8 + 1.5 \\ \\ \\ MN = \large \textbf{9.5 cm}$

Thus the length of MN is 9.5 cm.

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Shortcut!

→→→ The length of MN is the average of the lengths of AB and DC ←←←

Therefore,

$\displaystyle MN = \frac{AB + DC}{2} \\ \\ \\ MN = \frac{11 + 8}{2} \\ \\ \\ MN = \frac{19}{2} \\ \\ \\ MN = \large \textbf{9.5 cm}$