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Areas of Parallelograms and Triangles

Short Answer Type

51. Given two points A and B and a positive real number k. Find the locus of a point P such that ar(ΔPAB) = k.

Given: Two points A and B and a positive real number k.
To find: The locus of a point P such that ar(ΔPAB) = k.
Construction: Draw PM ⊥ AB.
Determination: Let PM = h

$space space space space space space space space space space space space space space space space space space space space space ar left parenthesis increment PAB right parenthesis equals straight k space space space space space space space space space space space space space space space space space space space space vertical line Given rightwards double arrow space space space space space space space space space space space space space space space 1 half left parenthesis AB right parenthesis left parenthesis PM right parenthesis equals straight k rightwards double arrow space space space space space space space space space space space space space space space space 1 half left parenthesis AB right parenthesis left parenthesis straight h right parenthesis equals straight k rightwards double arrow space space space space space space space space space space space space space space space space space space space space space straight h space equals space fraction numerator 2 straight k over denominator AB end fraction$

Given: Two points A and B and a positive real number k.To find: The l

∵ Points A and B are given.
∴ AB is fixed.
Also, k being a positive real number k is fixed. ∴ h is a fixed positive real number.
∴ The locus of P is a line parallel to the line
AB at a fixed distance

$Given: Two points A and B and a positive real number k.To find: The l$ $Given: Two points A and B and a positive real number k.To find: The l$ $Given: Two points A and B and a positive real number k.To find: The l$ $fraction numerator 2 straight k over denominator AB end fraction$

on either side of it.

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52. Show that the line segment joining the mid-points of a pair of opposite sides of a parallelogram divides it into two equal parallelograms.

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53. In an equilateral triangle, O is any point is the interior of the triangle and perpendiculars are drawn from O to the sides. Prove that the sum of these perpendicular line segments is constant.

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54. ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

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55. ΔABC and ΔABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ΔABC) = ar (ΔABD).

ar left parenthesis quad. space ABCD right parenthesis equals 1 half BD. left parenthesis AM plus CN right parenthesis.

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Long Answer Type

56.

ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove that

(i) ar(ΔADO) = ar(ΔCDO)
(ii) ar(ΔABP) = ar(ΔCBP)

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Short Answer Type

57. In the given figure, ΔABC is right angled at B, and BD is its median. E is the mid-point of BD. If AB = 6 cm, AC = 10 cm, calculate area of ΔBEC.

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Long Answer Type

58. In the given figure, ΔABC is right angled at A and AD is its median. BA is produced to E such that BA = AE. ED is joined. If AB = 6 cm, BC = 10 cm, find ar(ΔBED).

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59.

The medians of ΔABC intersect at G. Prove that

ar(ΔAGB) = ar(ΔAGC) = ar(ΔBGC) $equals 1 third space ar left parenthesis increment ABC right parenthesis$

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Short Answer Type

60. Triangles ABC and DBC are on the same base BC with vertices A and D on opposite sides of BC such that ar (ΔABC) = ar(ΔDBC). Show that BC bisects AD.

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