Outside the middle school program / Midpoint theorem 4 exercises (100% corrected)

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ABCDEIJF3cm CABDEOFG ABCDEIJF3cm4cm 1. Unclassified financial years E.4793 Consider the parallelogram ABCD shown below where AB =4 cm and I is the midpoint of [ AB ] . The point E is placed so that the point D is the midpoint of the segment [ AE ] . The point J is the point of intersection of the straight lines ( EI ) and ( CD ) . The point F is the point of intersection of the straight lines ( BC ) and ( EI ) . 1 Determine the measure of segment [ DJ ] . 2 a Determine the value of the quotient IB CJ . b Deduce the measure of segment [ BC ] . E.4679 In the plane, consider a triangle ABC whose dimensions are: AB = 6 cm ; AC = 9 cm ; ABC = 60 o Let us note I and J the two points of the half-line [ AB ) veri-fying the measures : AI = 2 cm ; AJ = 4 cm Note K and L the two points of the half-line [ AC ) verifying the measures : AK = 3 cm ; AL = 6 cm Note M the point of intersection of the straight lines ( JL ) and ( BK ) . 1 Make a drawing of this configuration. 2 Show that the straight lines ( IK ) and ( JL ) are parallel. 3 Justify that the point M is the middle of the segment [ BK ] . 4 Show that the line ( JL ) is parallel to the line ( BC ) . 5 Deduce that ( IK ) is parallel to ( BC ) . E.3672 Consider the figure below : Data from the above figure : CDE is a right triangle in C . A belongs to the segment [ CD ] , B belongs to the segment [ CE ] , and the line ( AB ) is parallel to the line ( DE ) . The point F is the middle of the segment [ AC ] and the point O is the middle of [ AB ] . The point G is the symmetric of F with respect to O . OF =12 cm ; AB =4.5 cm ; AC =1.8 cm The following questions are independent of each other : 1 What is the nature of the quadrilateral AFBG ? Justify. 2 Show that the line ( FO ) is parallel to the line ( CB ) . 3 Calculate the length CD . 4 Calculate a value approximating the angle to the nearest degree BAC . E.4792 Consider the parallelogram ABCD shown below : AB = 4 cm ; I is the middle of [ AB ] . The point E is the symmetric of the point A about the point D . The point J is the point of intersection of the lines ( EI ) and ( CD ) . The point F is the point of intersection of the lines ( BC ) and ( EI ) . 1 Determine the measure of segment [ DJ ] . 2 Determine the measure of segment [ BC ] . 3 Determine the measure of segment [ EJ ] . https://chingmath.fr chapExoCorrec/4793 sacados/4793 ABCDEIJF3cm chapExoCorrec/4679 sacados/4679 Generalisation du theoreme des milieux au partage d'un segment en trois parts. chapExoCorrec/3672 sacados/3672 CABDEOFG chapExoCorrec/4792 sacados/4792 ABCDEIJF3cm4cm