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51. ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that $CQ equals 1 fourth$ AC (see figure). If PQ produced meets BC at R, prove that R is the mid-point of BC.

Given: ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that

CQ equals 1 fourth

AC. PQ produced meets BC at R.
To Prove: R is the mid-point of BC.
Construction: Join BD to intersect AC at O.
Proof: ∵ ABCD is a parallelogram and the diagonals of a parallelogram bisect each that

$therefore space space space AO equals OC equals 1 half AC Now comma space space CQ equals 1 fourth AC space space space space space space space space space space space space space space equals 1 fourth left parenthesis 2 space OC right parenthesis space space space space space space space space space space space space space space equals space OC over 2$

$rightwards double arrow$ Q is the mid-point of CO
In ∆CDO,
∵ P is the mid-point of DC and Q is the midpoint of CO
∴ PQ || DO | by mid-point theorem
⇒ PR || DB
⇒ QR || OB
Now, in ∆COB,
∵ Q is the mid-point of CO and QR || OB
∴ R is the mid-point of BC
| by converse of mid-point theorem

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52. Prove that the line segment joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides.

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Short Answer Type

53. In ∆ABC, D, E and F are respectively the midpoints of sides AB, BC and CA (see figure). Show that ∆ABC is divided into four congruent triangles.

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54. I. m and n are three parallel lines intersected by transversals p and q such that I, m and n cut off equal intercepts AB and BC on p (see figure). Show that l, m and n cut off equal intercepts DE and EF on q also.

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55.

Points A and B are on the same side of a line l. AD and BE are perpendiculars to I, meeting at D and E respectively. C is the mid-point of AB. Prove that CD = CE
[Hint: From C draw the perpendicular CM on l.]

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56. In equilateral triangle ABC, the mid-points of the sides BC, CA and AB are respectively D, E and F as shown in the figure. Prove that DEF is also an equilateral triangle.

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57.

P is the mid-point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R (see figure). Prove that

(i) AR = 2 BC
(ii) BR = 2 BQ.

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58. In given figure ABCD is a trapezium in which AB || DC, BD is the diagonal and P is the mid- point of AD. A line is drawn through P, parallel to AB intersecting BC at Q. Show that CQ is equal to QB.

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59. In the figure, points A and B are on the same side of a line I. AD ⊥ l and BE ⊥ I at D and E respectively. If C is the mid-point of AB. prove that CD = CE.