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Diagonals of a Square are Equal in Length and they Meet at Right Angles

Here we will prove that in a square, the diagonals are equal in length and they meet at right angles.

Given: PQRS is a square in which PQ = QR = RS = SP, and ∠QPS = ∠PQR = ∠QRS = ∠RSP = 90°.

To prove: PR = QS and PR ⊥ QS

Proof:

            Statement

            Reason

1. In ∆SPQ and ∆RQP,

(i) SP = QR

(i) Given

(ii) PQ = PQ

(ii) Common side

(iii) ∠SPQ = ∠PQR

(iii) Given

(iv) ∆SPQ ≅ ∆RQP

Therefore, QS = PR    (Proved)

(iv) By SAS criterion of congruency. CPCTC.

2.

(v) ∠PQS = ∠PSQ

(v) In ∆PQS, PQ = PS

(vi) ∠PQS + ∠PSQ = 90°.

(vi) In ∆QPS, ∠QPS = 90° and sum of three angles of a triangle is 180°.

(vii) ∠PQS = 90°2 = 45°

(vii) By statements (v) and (vi).

(viii) ∠QPR = 45°

(viii) Similarly as (vi) and (vii) for the ∆PQR.

(ix) ∠POQ = 180° - (PQO + ∠QPO)

               = 180° - (45° + 45°)

               = 180° - 90°

               = 90°

Therefore, OP ⊥ OQ

Therefore, ∠POQ = 90°

Therefore, PR ⊥ QS       (Proved)

(ix) By statements (vii), (viii) and the sum of the angles of ∆POQ is 180°.






9th Grade Math

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