9.23, BD || CA, E is mid-point of CA and BD = 1/2 CA. 9.22, l, m, n, are straight lines such that l || m and n intersects l at P and m at Q. Points P and Q on BC trisects BC in three equal parts. If triangle ABC and DEF are congruent then. The opposite sides of a parallelogram are always parallel and equal in length. ![]() If the mid-points of the sides of a quadrilateral are joined in order, it is proven that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. Parallelogram: A parallelogram is a polygon formed by 4 points on a plane in which no three are collinear having four sides and corresponding four angles. NCERT Exemplar Class 9 Maths Exercise 9.3 Problem 9 If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. ☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 9 ID: A 1 G.CO.C.11: Quadrilateral Proofs Answer Section 1 ANS: 2 REF: 011411ge 2 ANS: Because ABCD is a parallelogram, AD CB and since ABE is a transversal, BAD and. Complete step-by-step answer: Let us start by drawing the diagram for a better visualisation of the situation given in the question. In a quadrilateral ABCD, the side AB is equal to the side DC. If AB = 6.2 cm, BC = 5.6 cm and CA = 4.6 cm, find the perimeter of trapezium FBCE If opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram. ✦ Try This: In Fig, D, E and F are the mid-points of the sides BC, CA and AB respectively of △ABC. Two parallel lines are equidistant throughout. Opposite sides of a parallelogram are equal. Opposite angles of a parallelogram are equal. Some informal degrees-of-freedom analysis: An arbitrary quadrilateral on a plane is described by 8 parameters: coordinates of each vertex (to simplify matter, I dont take quotient by isometries). If a diagonal is drawn in a parallelogram, then two congruent triangles are formed. Similarly, ar(CFP) = 1/4 ar(BCD) - (4)Īr(ASR) + ar(CFP) = 1/4 ar(BDA) + 1/4 ar(BCD)Īr(ASR) + ar(CFP) = 1/4 ar(BCDA) - (5) A quadrilateral with both pairs of opposite sides parallel. We know that the median of a triangle divides it into two triangles of equal areas. P, F, R and S are the midpoints of the sides BC, CD, AD and AB Thus the quadrilateral ABCD shown opposite is a parallelogram because AB DC and DA. We could have also done this by drawing the second diagonal DB, and used the two triangles ΔADB and ΔCDB instead.If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. A parallelogram is a quadrilateral whose opposite sides are parallel. ![]() ![]() Since PQ and SR are both parallel to a third line (AC) they are parallel to each other, and we have a quadrilateral (PQRS) with two opposite sides that are parallel and equal, so it is a parallelogram. 1 Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its opposite angles are congruent. Buildings: Many buildings are constructed, keeping in mind the shape of parallelograms.A famous real-life illustration is the Dockland Office Building in Hamburg, Germany. So, using the Triangle Midsegment Theorem we find that PQ||AC and PQ = ❚C, and also that SR||AC and SR = ❚C. When we look around us, we can see multiple parallelogram-like shapes and objects in the form of buildings, tiles, or paper. We now have two triangles, ΔBAC and ΔDAC, where PQ and SR are midsegments. Since ADB DBC and the sum of the angles of a triangle is 180°, Therefore, A and ABC are supplementary. We have no triangles here, so let's construct them, so the midpoints of the quadrilateral become midpoints of triangles, by drawing the diagonal AC: The fact that we are told that P, Q, R and S are the midpoints should remind us of the Triangle Midsegment Theorem - the midsegment is parallel to the third side, and its length is equal to half the length of the third side. In a quadrilateral ABCD, the points P, Q, R and S are the midpoints of sides AB, BC, CD and DA, respectively. 3 Given: Quadrilateral ABCD, diagonal AFEC, AE FC, BF AC, DE AC, 1 2 Prove: ABCD is a parallelogram. ![]() Surprisingly, this is true whether it is a special kind of quadrilateral like a parallelogram or kite or trapezoid, or just any arbitrary simple convex quadrilateral with no parallel or equal sides. If you connect the midpoints of the sides of any quadrilateral, the resulting quadrilateral is always a parallelogram.
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