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Ex 8.1, 5 - Show that if diagonals of a quadrilateral are equal - Diagonal of parallelogram

Ex 8.1, 5 - Chapter 8 Class 9 Quadrilaterals - Part 2
Ex 8.1, 5 - Chapter 8 Class 9 Quadrilaterals - Part 3
Ex 8.1, 5 - Chapter 8 Class 9 Quadrilaterals - Part 4

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Ex 8.1, 5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. Given: Let ABCD be the quadrilateral. Diagonals are equal, i.e., AC = BD & bisect each other, i.e. OA = OC & OB = OD, at right angles ,i.e., ∠AOB = ∠BOC = ∠COD = ∠AOD = 90° To prove: ABCD is a square Proof: Square is a parallelogram with all sides equal and one angle 90° First we will prove ABCD is a parallelogram and then prove all sides equal , and one angle equal to 90° In ΔAOB and ΔCOB, OA = OC ∠AOB = ∠COB OB = OB ∴ ΔAOB ≅ ΔCOB ∴ AB = CB Similarly we can prove ΔAOB ≅ ΔDOA , so AB = AD & ΔBOC ≅ ΔCOD , so CB = DC So, AB = AD = CB = DC Now we can say that AB = CD & AD = BC In ABCD, both pairs of opposite sides are equal, Hence, ABCD is a parallelogram Square is a parallelogram with all sides equal and one angle 90° So, we prove one angle 90° In ΔABC and ΔDCB, AC = BD AB = DC BC = CB ∴ ΔABC ≅ ΔDCB ⇒∠ ABC = ∠ DCB Now, AB ∥ CD & BC is transversal ∠ B + ∠ C = 180° ∠ B + ∠ B = 180° 2∠ B = 180° ∠ B = (180°)/2 = 90° Thus, ABCD is a parallelogram with all sides equal and one angle 90° So, ABCD is a square

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.