Question

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.​

Answer 1

Answer:

Hii..

${\huge{\tt{\boxed{\fcolorbox{red}{pink}{Solution:-}}}}}$

_____________________________________

Given:

Let's suppose AB and CD are the two chords of a circle intersecting at Z. PQ is the diameter which passes through their point of intersection as well. The chords are equally inclined to the diameter. It implies that

∠OLE = ∠OME.

To prove the length of chord AB and CD are equal.

-------------------------

Proof:

Construction: Drop two perpendiculars OL and OM from center to the chord AB and CD respectively, such that they intersect the chords at Land M respectively,

In triangle OLE and triangle OME ∠OLE = ∠OME (Both are equal to 90°, as per our construction)

∠LEO = ∠MEO(Chords are equally inclined with the diameter)

OE = OE (Both are common in the two = triangles.)

Hence, ∆OLE ≈ ∆OME by AAS(Angle-Angle-Side) criteria

Thus, it can be concluded that

OL=OM (By CPCT- Corresponding Parts of Congruent Triangle)

Since the chords are equidistant from the center of the circle, therefore they are equal to each other.

Thus, length of chord AB = length of chord CD (Proved)

-----------------------------

Note:-

The two triangles are said to be congruent if their corresponding sides and corresponding angles are equal.

There are 4 ways by which it can be proved SSS (Side-Side-Side) Criteria

SAS (Side-Angle-Side)

AAS (Angle-Angle-Side) ASA (Angle-Side-Angle)

AAA is not a criterion to prove the congruency of the triangle.

_____________________________________

Hope it helps you dear!