Theorem 10.4 Class 9 Maths Explanation with Proof

Today, we will discuss theorem 10.4 class 9 Maths which is related to Chapter 10 Circles Class 9 Mathematics. After understanding theorem 10.4, you can solve the exercise questions given in the NCERT book of Class 9 Maths.

 Theorem 10.4 Class 9

The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Given 

 AB is a chord of a circle with centre O. A line OM is drawn through the centre O to chord AB such that M is the mid-point of AB, i.e. AM=MB.

To Prove 

 OM⊥AB

Construction 

Join OA and OB.

Proof 

In ∆OAM and ∆OBM, we have

AM=MB               (∵M is the midpoint of AB)

OA=OB               (Radii)

OM=OM              (common sides)

∴∆OAM=∆OBM (By SSS congruence)

⟹ ∠OMA=∠OMB (By CPCT) …….(i)

Now AB is a straight line

⟹ ∠OMA+∠OMB=180° (Linear pair)

⟹ 2∠OMB=180° (Using equation –(i))

⟹ `∠OMB= frac{180°}{2}` 

⟹ OMA=90° 

Thus     ∠OMA=∠OMB=90° 

And       OM⊥AB    Hence proved

Related Topics

1. Theorem 10.5

2. Theorem 10.3