Consider △ ABC We know that E and F are the midpoints Based on the midpoint theorem We know that EF || AC and EF = ½ AC Consider △ ADC We know that H and G are the midpoints Based on the midpoint theorem We know that HG || AC and HG = ½ AC So we get EF || HG and EF = HG = ½ AC ……….. (1) Consider △ BAD We know that H and E are the midpoints Based on the midpoint theorem We know that HE || BD and HE = ½ BD Consider △ BCD We know that G and F are the midpoints Based on the midpoint theorem We know that GF || BD and GF = ½ BD So we get HE || GF and HE = GF = ½ BD …….. (2) We know that the diagonals of a square are equal So we get AC = BD …….. (3) By using equations (1), (2) and (3) So we get GF || BD and HE || GF We have EF = GH = GH = HE We know that EFGH is a rhombus From the figure, we know that the diagonals of a square are equal and intersect at right angles ∠ DOC = 90o We know that the sum of adjacent angles of a parallelogram is 180 It can be written as ∠ DOC + ∠ GKO = 180 By substituting the values 90 + ∠ GKO = 180 On further calculation ∠ GKO = 180 - 90 By subtraction ∠ GKO = 90 From the figure, we know that ∠ GKO and ∠ EFG are corresponding angles We get ∠ GKO = ∠ EFG = 90 We know that ∠ EFG = 90 Hence, EFGH is a square. Therefore, it is proved that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a square is a square. Geometry is derived from a Greek word that means ‘Earth Measurement’. It is a branch of mathematics and is concerned with the properties of space i.e., a visual study of shapes, position of figures, patterns, sizes, etc. Geometry is a subject of countless developments, so there exist many types. They are Euclidean Geometry, Non-Euclidean Geometry, Algebraic Geometry, Riemannian Geometry, and Symplectic Geometry. QuadrilateralA quadrilateral can be separated into two words Quad means four and lateral means side. So a quadrilateral is a closed figure with four sides. It had four vertices. The sides of a quadrilateral are equal/ unequal/ parallel/ irregular which leads to various geometric figures. Example: Square, Rectangle, Rhombus, Parallelogram, Trapezium, etc. The midpoint of a side divides a side of any figure into two equal parts (length-wise). In a quadrilateral, there will be a midpoint for each side i.e., Four mid-points. There are a few factors that determine the shape formed by joining the midpoints of a quadrilateral. Those factors are the kind of quadrilateral, diagonal properties, etc. These factors affect the shape formed by joining the midpoints in a given quadrilateral. Let’s look into the various scenarios to get a better understanding. Solution:
Sample QuestionsQuestion 1: Consider the rhombus ABCD which is also a kind of Quadrilateral. Find the shape of the figure formed by joining the midpoints. Solution:
Question 2: If the figure formed by joining the midpoint of a quadrilateral is square only if, do explain the condition. Solution:
Question 3: What is the figure formed by joining the midpoints of a parallelogram. Solution:
Question 4: What is the figure formed by joining the midpoints of a quadrilateral whose diagonals are of length equal? Answer:
Question 5: What is the figure formed by joining the midpoints of a quadrilateral whose diagonals are perpendicular but not of equal length? Answer:
Figure can be drawn as: Let ABCD be a quadrilateral such that P,Q,R and S are the mid-points of side AB,BC,CD and DA respectively. In ΔABC,P and Q are the mid-points of AB and BC respectively. Therefore, PQ || AC and `PQ = 1/2 AC` Similarly, we have RS || ACand `RS = 1/2 AC` Thus, PQ || RS and PQ = RS Therefore, PQRSis a parallelogram. The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a parallelogram. Hence the correct choice is (a). Open in App Suggest Corrections 2 |