Question

‘I’ is the perpendicular bisector of side BC of $\vartriangle ABC$ meeting the circumference of the triangle of M opposite to A then
(a)$\angle MBC=\angle MCB$
(b) $\angle BCM=\angle BAM$
(c) $\angle CMB=\angle CAM$
(d) $\angle BAM=\angle CAM$

Answer 1

Hint: We start solving the problem by drawing the figure representing the given information. We then use the property that an arc in a circle subtends equal angles anywhere on the circumference. We then check whether both angles given in each option are satisfying this property to get the correct option.

Complete step-by-step solution:
Let us draw the figure representing the given information.

From the property of angle subtended by an arc, we know that an arc in a circle subtends equal angles anywhere on the circumference.
In this figure, BM is the chord.
$\angle BAM\And \angle BCM$ is subtended at the end of this chord.
Hence, by this property,
$\angle BCM=\angle BAM$
Hence, option (b) is correct.
Now, we will start checking other options.
(A) $\angle MBC=\angle MCB$
Here, look at the angles $\angle MBC\And \angle MCB$, we can see that the $\angle MBC$ is subtended at the arc MC and the angle $\angle MCB$ is subtended at the arc BM.
For the property to get applied, the angles should be subtended at the same arc.
Therefore, option (A) is incorrect.
(C) $\angle CMB=\angle CAM$
Here, look at the angles $\angle CMB\And \angle CAM$, we can see that the $\angle CMB$ is subtended at the arc BC and the angle $\angle CAM$is subtended at the arc CM.
For the property to get applied, the angles should be subtended at the same arc.
Therefore, option (C) is incorrect.
(D) $\angle BAM=\angle CAM$
Here, look at the angles $\angle BAM\And \angle CAM$, we can see that the $\angle BAM$ is subtended at the arc BM and the angle $\angle CAM$is subtended at the arc CM.
For the property to get applied, the angles should be subtended at the same arc.
Therefore, option (D) is incorrect.
Hence option B is the correct answer

Note: We can also find all the angles required by assuming one angle in the triangle and finding the other angles. We can see that the triangle ABC is an isosceles triangle with equal sides AB and AC. We can use the fact that the angles to equal sides in a triangle are equal and the sum of the angles in a triangle is ${{180}^{\circ }}$ to get the remaining angles. We can also see that the triangle MBC is also an isosceles triangle with equal sides MB and MC to get the other angles.