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

Our aim is to prove that the diagonals of a Square are Equal in Length & they Meet at Right Angles. The theorems shown on this page are prepared by the math experts. Hence refer to our articles to score good marks in the exams. The solution to the theorem is yes, the diagonals of the square are equal and they bisect each other at the right angles.

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

Statement:
Show that the diagonals of a square are equal and bisect each other at right angles?
Given:
Diagonals of a square in equal length bisect at the right angles
PQRS is a square in which PQ = QR = RS = SP, and ∠QPS = ∠PQR = ∠QRS = ∠RSP = 90°.
To prove:
PR = QS and PR ⊥ QS
Proof:
1. In ∆SPQ and ∆RQP,
Given
SP = QR….. equation 1
(ii) PQ = PQ is a common side…equation 2
Given ∠SPQ = ∠PQR….. equation 3
Therefore ∆SPQ ≅ ∆RQP…. equation 4
Therefore, QS = PR [ by SAS criterion of congruency CPCTC]
Hence proved
2. ∠PQS = ∠PSQ….. equation 5
In ∆PQS, PQ = PS
∠PQS + ∠PSQ = 90°……. equation 6
In ∆QPS, ∠QPS = 90° and sum of three angles of a triangle is 180°.
∠PQS = 90°/2 = 45°
By using equation 5 and 6
∠QPR = 45°…. equation 7
Similarly as (6) and (7) for the ∆PQR.
∠POQ = 180° – (PQO + ∠QPO)
= 180° – (45° + 45°)
= 180° – 90°
= 90°
Therefore, OP ⊥ OQ
Therefore, ∠POQ = 90°
Therefore, PR ⊥ QS
Hence proved.

FAQs on Diagonals of a Square are Equal in Length & Bisect Each Other at Right Angles

1. Do the diagonals of a square meet at right angles?

The diagonals of a square bisect each other and meet at 90°. The diagonals of a square bisect its angles. All the four angles of a square are equal so that 360°/4 = 90°. It is a right angle.

2. Is the diagonals of a square bisect at right angles?

The diagonals of a square are equal in length and the diagonals of a square bisect each other and the diagonals of a square bisect each other at right angles.

3. How do you prove that the diagonals of a square are equal?

Let the diagonals AC and BD intersect each other at a point O. To prove that the diagonals of a square are equal and bisect each other at right angles, we have to prove AC = BD, OA = OC, OB = OD, and AOB = 90º.
In ABC and DCB,
AB = DC (Sides of a square are equal to each other)
ABC = DCB (All interior angles are of 90)
BC = CB (Common side)
ABC = DCB (By SAS congruency)
AC = DB (By CPCT)
the diagonals of a square are equal in length.

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