Show that the diagonals of a square are equal and bisect each other at right angles
Solution:
Given: The quadrilateral is a square.
To prove: Diagonals of a square are equal and bisect each other at right angles.
Thus, we have to prove AC = BD, OA = OC, OB = OD, and ∠AOB = 90°
Let ABCD be a square.
Let the diagonals AC and BD intersect each other at a point O.
In ΔABC and ΔDCB,
AB = DC (Sides of a square are equal to each other)
∠ABC = ∠DCB (All interior angles are of 90o)
BC = CB (Common side)
∴ ΔABC ≅ ΔDCB (By SAS congruence rule)
∴ AC = DB (By CPCT) ------------ (1)
Hence, the diagonals of a square are equal in length.
In ΔAOB and ΔCOD,
∠AOB = ∠COD (Vertically opposite angles)
∠ABO = ∠CDO (Alternate interior angles)
AB = CD (Sides of a square are always equal)
∴ ΔAOB ≅ ΔCOD (By AAS congruence rule)
∴ AO = CO and OB = OD (By CPCT) ------------ (2)
Hence, the diagonals of a square bisect each other.
In ΔAOB and ΔCOB,
As we had proved that diagonals bisect each other,
Therefore, AO = CO
AB = CB (Sides of a square are equal)
BO = BO (Common)
∴ ΔAOB ≅ ΔCOB (By SSS congruency)
∴ ∠AOB = ∠COB (By CPCT) --------- (3)
However, ∠AOB + ∠COB = 180° (Linear pair)
2∠AOB = 180° [From equation (3)]
∠AOB = 90° ------------- (4)
Hence, from equation (1), (2) and (4), we see that the diagonals of a square are equal and bisect each other at right angles
☛ Check: NCERT Solutions for Class 9 Maths Chapter 8
Video Solution:
Show that the diagonals of a square are equal and bisect each other at right angles.
NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 4
Summary:
Thus, we have proved that the diagonals of a square are equal and bisect each other at right angles.
☛ Related Questions:
- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
- Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus.
- ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
- ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:(i) ABCD is a square(ii) diagonal BD bisects ∠B as well as ∠D.