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 = \(\frac{90°}{2}\) = 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°. |
From Diagonals of a Square are Equal in Length & they Meet at Right Angles to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.