Concept:
- Congruence triangle: If two or more triangles are congruent, then all of their corresponding angles and sides are as well congruent.
- 'Side-Angle-Side' triangle congruence theorem: If the two sides of a triangle are equal to the two sides of another triangle, and the angle formed by these sides in the two triangles are equal, then these two triangles are congruent.
- Concentric Circles: Concentric circles are circles with a common center.
- The perpendicular from the center of a circle to a chord bisects the chord.
Calculation:
Statement: 1
Given that,
AB and
AC are two equal chords of the circle.
In
△APB and
△APC ,
⇒AB=AC [Given
] ⇒∠BAP=∠CAP[ Given
] ⇒AP=AP [Common]
⇒△APB≅△APC [by SAS congruence criterion]
⇒ So, by property of congruence triangle we have
⇒BP=CP and
∠APB=∠APC ⇒∠APB+∠APC=180∘ [Linear pair
] ⇒2∠APB=180∘[∠APB=∠APC] ⇒∠APB=90∘ Now,
BP=CP and
∠APB=90∘ Therefore,
AP is the perpendicular bisector of chord
BC .
Hence, AP passes through the center
O of the circle.
Statement: 2
Draw a perpendicular from the center of the circle
OM to the line
AD .
We can see that
BC and
AD are the chords of the smaller and bigger circle respectively.
We know that a perpendicular drawn from the center of the circle bisects the chord.
⇒BM=MC⋯(1) ⇒AM=MD⋯−(2) Subtracting (2) from (1), we get
AM−BM=DM−CM ⇒AB=CD ⇒AB+BC=CD+bC ⇒AC=BD ∴ Both 1 and 2 are correct.