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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

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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 $AF equals 1 third AC$

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

To Prove: AF=

1 third

AC
Construction: Draw DG || EF to intersect AC in G.

Given: In ∆ABC, AD is the median through A and E is the mid-point o

Proof: ∵ EF || DG    | by construction
and    AE = ED
| ∵ E is the mid-point of AD
∴ AF = FG    ...(1)
| A line drawn through the mid-point of a side of a triangle parallel to another side bisects the third side
Again,
DG || EF    | by construction
and    CD = DB
| ∵ D is the mid-point of
∴ BC FG = GC    ...(2)
| A line drawn through the mid-point of a side of a triangle parallel to another side bisects the third side
From (1) and (2),
AF = FG = GC
Now, AF + FG + GC = AC
⇒ AF + AF + AF = AC
| From (1) and (2)
⇒ 3 AF = AC

rightwards double arrow space space 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 =

1 half

(AB + DC).

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