![]() Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. A villager Itwaari has a plot of land of the shape of a quadrilateral.Show that i) ar (ACB) = ar (ACF) ii) ar (AEDF) = ar (ABCDE) A line through B parallel to AC meets DC produced at F. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O.If side AB of a parallelogram ABCD is produced to any point P, a line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed, then ar (ABCD) = ar (PBQR). Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.3 Question 9 A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Fig. Video Solution: The side AB of a parallelogram ABCD is produced to any point P. ☛ Check: NCERT Solutions Class 9 Maths Chapter 9 Since AC and PQ are diagonals of parallelograms ABCD and PBQR respectively, ΔACQ and ΔAQP are lying on the same base AQ and existing between the same parallels AQ and CP.Īccording to Theorem 9.2: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.Īr (ΔACQ) - ar (ΔABQ) = ar (ΔAPQ) - ar (ΔABQ) Position vector of point D w.The side AB of a parallelogram ABCD is produced to any point P. Position vector of point B w.r,t the origin O, #vec(OB)= X_Bhati +Y_Bhatj# Position vector of point A w.r,t the origin O, #vec(OA)= X_Ahati +Y_Ahatj# It can also be established considering #vec(AB)# and # vec(AD)# This formula will give the area of the parallelogram. Since diagonal BD bisects the parallelogram into two congruent triangle. We can easily establish the formula for calculation of the area of a parallelogram ABCD, when any three vertices (say A,B,D) are known. ![]() In terms of original coordinates, it looks like this: Now we know all components to calculate the area:Īltitude #|DH|=sqrt(U_D^2+V_D^2)*|U_A*V_D-V_A*U_D| / sqrt# ![]() The length #AD# is the length of vector #q#:Īngle #/_BAD# can be determined by using two expressions for the scalar (dot) product of vectors #p# and #q#: The length of altitude #|DH|# can be expressed as #|AD|*sin(/_BAD)#. Our parallelogram now is defined by two vectors:ĭetermine the length of base #AB# as the length of vector #p#: ![]() Then the ( #U,V#) coordinates of all vertices will be: So, we will perform the following transformation of coordinates: The area will be the same, but calculations will be easier. To determine the area of our parallelogram, we need the length of its base #|AB|# and the altitude #|DH|# from vertex #D# to point #H# on side #AB# (that is, #DH_|_AB#).įirst of all, to simplify the task, let's move it to a position when its vertex #A# coincides with the origin of coordinates. Let's assume that our parallelogram #ABCD# is defined by the coordinates of its four vertices - #, #, #, #. ![]()
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