Class 9 Mathematics · Ganita Manjari Part 1

Class 9 Maths Chapter 5 – Exercise Set 5.6 Solutions

Question 1 — Chord from a central angle

Given: $OA=OB=12$ cm and $\angle AOB=60^\circ$.

Since $OA=OB$, $\triangle AOB$ is isosceles. Its two remaining angles are equal:

Thus the triangle is equilateral.

Answer: $\boxed{AB=12\text{ cm}}$.

Question 2 — Angles subtended by the same chord

(i) Same side of chord

If $X$ and $Y$ lie on the same side of chord $AB$, they see the same chord in the same segment. Therefore the subtended angles are equal:

So two such points cannot give different angles.

(ii) Equal angles do not force the points to be on the same side

No. Equal angles subtended by the same chord can occur at points on different arcs/sides in the circle. Therefore equality of the angles alone does not imply that $X$ and $Y$ are on the same side.

(iii) Equal angles at two exterior points

When the equal-angle condition is taken in the appropriate same-side configuration, the converse of the concyclicity theorem applies. Thus the four points are concyclic, so the circle through $A,B,X$ also passes through $Y$.

Question 3 — Find the unknown angle in the given cyclic figure

The four relevant points lie on the same circle, so the quadrilateral is cyclic. The angle opposite the given $100^\circ$ angle is $x$.

Answer: $x=80^\circ$.