Exercise-9.1, Class 9th, Maths, Chapter 9, NCERT

1.

Statement. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Proof (simple and direct).
Let the two congruent circles have centres $O_1$ and $O_2$ and equal radii $r$. Let $AB$ be a chord of the first circle and $CD$ be a chord of the second circle with $AB=CD$. Join $O_{1}A,O_{1}B$ and $O_{2}C,O_{2}D$

In triangle $O_{1}AB$ and triangle $O_{2}CD$:

• $O_{1}A=O_{1}B=r$ and $O_{2}C=O_{2}D=r$ (radii),

• $AB=CD$ (given).

Thus each triangle is isosceles with equal legs and equal base. Concretely, by SSS (since $O_{1}A=O_{2}C,O_{1}B=O_{2}D$ and $AB=CD$ after matching corresponding sides between the two triangles), the two triangles $O_{1}AB$ and $O_{2}CD$ are congruent. Therefore the angles at the centres that subtend the chords are equal:

$$\mathrm{\angle }A{O}_{1}B=\mathrm{\angle }C{O}_{2}D\mathrm{}$$

So equal chords in congruent circles subtend equal central angles. ■

(Geometric idea: identical radii + equal chord ⇒ the isosceles triangles formed by the radii and chord are congruent, hence equal central angles.)

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Q2.

Statement. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Proof (the converse).
Take two congruent circles with centres $O_1$ and $O_2$ (same radius $r$). Let chords $AB$ and $CD$ subtend equal central angles:

$$\mathrm{\angle }A{O}_{1}B=\mathrm{\angle }C{O}_{2}D\mathrm{}$$

Consider the isosceles triangles $O_{1}AB$ and $O_{2}CD$. Each has two sides equal to the radius $r$. The included angles at the centres are equal (hypothesis). So the two triangles are congruent by SAS (side–angle–side: two radii and the included central angle). Hence corresponding bases are equal:

$$AB=CD$$

$$\mathrm{}$$So equal central angles in congruent circles imply equal chords. ■

(Geometric idea: equal radii + equal central angle ⇒ congruent isosceles triangles ⇒ equal chord lengths.)