Theorem: In a square the diagonals are equal and perpendicular to each other.
Given: A square ABCD. To Prove: AC = BD and AC and BC intersect perpendicularly. Proof: In ADB and BCA AD = BC [ Sides of square are equal] DAB = CBA [Each angle is ] AB = AB [ Common] Therfore DAB CBA [SAS Criteria of Congruence] BD = AC As ABCD is a square Therefore it is a parallelogram. Also diagonals of a parallelogram bisect each other. AO = OC and BO = OD ...................(1) In BOC and COD BO = DO [ From Equation (1)] BC = CD [ Sides of a square] OC = OC [ Common ] Therfore BOC DOC [SSS Criteria of Congruence] [ By CPCT ] ............(2) But [Linear Pair] [Using Eq 2]
Similarly we can prove that Hence Proved that In a square the diagonals are equal and perpendicular to each other. | |
Theorem : If the diagonals of a parallelogram are equal and intersect at right angles, then the parallelogram is a square. | |
Given : ABCD is a parallelogram, such that AC = BD and AC BD To Prove: ABCD is a square Proof: In AOB and COB AO = OC [ Diagonals of parallelogram bisect each other] BO = OB [ Common] AOB = COB [ Each as AC BD] Therefore AOB COB [ SAS Criteria of Congurence] AB = BC [CPCT] As ABCD is a IIgm and the adjacent sides are equal AB = BC Therefore AB = BC = CD= DA. In ADB and BCA AD = BC [ Proved above] BD = AC [Given] AB = AB [ Common] Therfore DAB CBA [SSS Criteria of Congruence] --------(1) [ CPCT] As ABCD is a parallelogram, AD || BC Now AD || BC and AB is the transversal [ Co interior angles are supplementary ] [ Replacing B from equation 1] Now ABCD is a IIgm in which all sides are equal and one angle is .Thus ABCD is a square |
Illustration: If the side of a square ABCD whose diagonals intersect at O is 6cm find the length of OA and OB;
Solution: As the diagonals of a square are equal and bisect each other perpendicularly We can say that OA = OB = x(say) ..............(1) and AOB = AB = 6cm ...................(2) [Given] Now AOB is aright angled at O [By Pythagorous Theorem] Substituting the values from Eq (1) and (2), we get cm Hence OA = OB = cm |
In the square PQRS, the diagonals bisect each other at O. Identify the type of triangle POQ. | |||
Right Option : D | |||
View Explanation |
If the side of a square ABCD whose diagonals intersect at O is cm find the length of OA and OB. | |||
Right Option : A | |||
View Explanation |
If the side of a square ABCD whose diagonals intersect at O is 8 cm find the length of OA and OB. | |||
Right Option : C | |||
View Explanation |
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