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Quadrilaterals

Long Answer Type

41. In a parallelogram ABCD, E and Fare the mid-points of sides AB and CD respectively. (see figure). Show that the line segments AF and EC trisect the diagonal BD.

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42. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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43.

ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that:

(i) D is the mid-point of AC (ii) MD ⊥ AC (iii) $CM space equals space MA space equals space 1 half AB.$

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Short Answer Type

44. In triangle ABC, points M and N on sides AB and AC respectively are taken so that

$AM equals 1 fourth AB space and space AN equals 1 fourth AC comma space$ Prove that MN = $1 fourth$

Given: In triangle ABC, points M and N on the sides AB and AC respectively are taken so that

AM equals 1 fourth AB thin space and space AN equals 1 fourth AC

To Prove:

MN equals 1 fourth BC.

Given: In triangle ABC, points M and N on the sides AB and AC respect

Construction: Join EF where E and F are the middle points of AB and AC respectively.
Proof: Y E is the mid-point of AB and F is the mid-point of AC.

$therefore space EF space parallel to space BC space space space and space space space EF space equals space 1 half BC space space space space space space space.... left parenthesis 1 right parenthesis Now comma space space space space space space space space space space space space space space space space space space AE equals 1 half AB and space space space space space space space space space space space space space space space space space space space space AM space equals space 1 fourth AB therefore space space space space space space space space space space space space space space space space space space space space space space AM equals 1 half space AE Similarly comma space space space space space space space space space space space space space AN equals 1 half AF$
$rightwards double arrow$ M and N are the mid-points of AE and AF respectively.

$therefore space MN space parallel to space EF space and space MN space equals space 1 half space EF equals 1 half open parentheses 1 half BC close parentheses$
| From (1)
$equals 1 fourth BC.$

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45. In triangle ABC, points M and N on sides AB and AC respectively are taken so that

AM equals 1 fourth AB space and space AN equals 1 fourth AC comma space Prove space that space MN equals 1 fourth

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46. ABCD is a rhombus and AB is produced to E and F such that AE = AB = BF. Prove that ED and FC are perpendicular to each other.

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Long Answer Type

47. In ∆ABC, AD is the median through A and E is the mid-point of AD. BE is produced to meet AC in F. Prove that

AF equals 1 third AC

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48.

D, E, F are respectively the midpoints of the sides BC, CA and AB of a triangle ABC. Show that:
(i) BDEF is a parallelogram
(ii) DFEC is a parallelogram.

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Short Answer Type

49. In the figure ABCD is a parallelogram and E is the mid-point of side BC. DE and AB on producing meet at F. Prove that AF = 2AB.

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Long Answer Type

50. ABCD is a trapezium in which side AB is parallel to the side DC and E is the mid-point of side AD (see figure). If F is a point on the side BC such that the segment EF is parallel to the side DC, prove that F is the mid-point of BC and EF =