Question

If the angle subtended by the minor arc BD at the centre is ${125}^{\circ }$, find the angle subtended by the minor arc CD at centre will be
 

A. ${35}^{\circ }$
B. ${70}^{\circ }$
C. ${125}^{\circ }$
D. ${150}^{\circ }$

Answer 1

Hint: In the question, it is given that the two chords BD and CD are of 8 units in length and angle $\mathrm{\angle }BAD$ is equal to ${125}^{\circ }$ and we are asked to find the angle $\mathrm{\angle }CAD$ . To solve this question, we have to see the relation between the triangles $\mathrm{\Delta }BAD\text{ and }\mathrm{\Delta }CAD$ and find the relation between the angles $\mathrm{\angle }BAD\text{ and }\mathrm{\angle }\text{CAD}$ to get the value of angle $\mathrm{\angle }CAD$. The two triangles $\mathrm{\Delta }BAD\text{ and }\mathrm{\Delta }CAD$ are congruent because of the SSS property of the congruence of triangles which states that two triangles are congruent if their corresponding sides are equal.

Complete step-by-step answer:

Consider the two triangles $\mathrm{\Delta }BAD\text{ and }\mathrm{\Delta }CAD$. In the question, it is given that the length of the sides
BD = 8 units.
CD = 8 units.
As seen in the above figures, the point A is the centre of the circle. The points B, C, D lie on the circle of radius of r units. So,
$AB=AC=AD=r\text{ units}$
Comparing the sides of the two triangles $\mathrm{\Delta }BAD\text{ and }\mathrm{\Delta }CAD$, we get
$BD=CD=8$ Unit (given in the question)
$AB=AC=r$ Unit (radius of the circle)
AD is the common side of the two triangles. So,
From the SSS (Side, Side, Side) property of the congruency of triangles which states that two triangles are congruent if their corresponding sides are equal, we can conclude that the triangles $\mathrm{\Delta }BAD\text{ and }\mathrm{\Delta }CAD$ are congruent.
In the congruent triangles the angles opposite to the corresponding sides are equal.
$\mathrm{\angle }BAD$ is the corresponding angle to the side BD and $\mathrm{\angle }CAD$ is the corresponding angle to the side CD. The sides BD and CD are corresponding sides to each other as they are the equal sides of two triangles.
$BD=CD\iff \mathrm{\angle }BAD=\mathrm{\angle }CAD$
$\therefore \mathrm{\angle }CAD=\mathrm{\angle }BAD={125}^{\circ }$.
Option C is the right answer.


Note:The two chords are meeting at the same point D in the given question. But even if there is no common point for the two chords, if the lengths of chords are equal, the angles subtended by the chords at the centre are equal. This is a property which will help in many questions of similar kind.