Mathematics · Class 9 · Constructions · Term 3

Construction of Specific Angles

Construct angles of 60°, 90°, 45°, 30°, and 22.5° using only a compass and straightedge, and understand the logic behind each construction.

TL;DR:Let's explore the hidden rules that govern the chords of a circle. We will discover the predictable and elegant relationship they share with the circle's centre.

• CBSE Learning Outcomes
• NCERT Class 9 Mathematics: Chapter 11 - Constructions

About This Topic

This topic delves into the fundamental properties of chords in a circle, a crucial area within Class 9 geometry as prescribed by the NCERT framework. Students move from basic definitions of circles to understanding the intricate relationships between a circle's components. The core of this chapter revolves around a few key theorems: the perpendicular from the centre to a chord bisects the chord, and its converse. Additionally, students will explore and prove that equal chords are equidistant from the centre, and its converse.

Understanding these theorems is not merely about memorisation; it is about developing deductive reasoning and proof-writing skills. These concepts form the bedrock for more advanced topics in circles in Class 10, such as tangents and cyclic quadrilaterals. The focus should be on visualising these properties through constructions, verifying them through activities, and then formalising the understanding with logical proofs, primarily using triangle congruence criteria (RHS, SSS) and the Pythagoras theorem.

Key Questions

1. Explain why the basic construction using two arcs from a line creates a 60° angle.
2. Justify the sequence of steps required to construct a 45° angle.
3. Analyse how the construction of a 90° angle relies on creating a perpendicular bisector.

Learning Objectives

• Prove that the perpendicular from the centre of a circle to a chord bisects the chord.
• Demonstrate that equal chords of a circle are equidistant from the centre.
• Apply the properties of chords to solve numerical problems involving lengths and distances.
• Construct the centre of a circle given an arc or three non-collinear points.
• Analyse the relationship between the length of a chord and its distance from the centre.

Key Vocabulary

Chord	A line segment whose endpoints both lie on the circle.
Centre	The fixed point inside a circle from which all points on the circle are equidistant.
Radius	A line segment from the centre of a circle to any point on the circle.
Equidistant	At an equal distance from a point or line.
Perpendicular Bisector	A line that divides another line segment into two equal parts at a right angle (90 degrees).

Watch Out for These Misconceptions

Common Misconception

Any line drawn from the centre to a chord will bisect it.

What to Teach Instead

Only the line drawn *perpendicular* from the centre to a chord bisects it. A non-perpendicular line (an oblique line) from the centre will not divide the chord into two equal halves.

Common Misconception

The distance from the centre to a chord is the length of the radius.

What to Teach Instead

The radius is the distance from the centre to a point *on the circle*. The distance to a chord is the length of the perpendicular segment from the centre *to the chord* itself. This distance is always less than or equal to the radius.

Common Misconception

All chords are smaller than the diameter.

What to Teach Instead

This is almost correct, but the diameter is itself the longest possible chord in a circle. It is a special chord that passes through the centre.

Active Learning Ideas

Experiential Learning

Paper Folding Verification

Students draw a circle on a piece of paper and draw a chord. They then fold the paper such that the endpoints of the chord coincide, creating a crease. This crease will pass through the centre and be perpendicular to the chord, visually demonstrating that the perpendicular from the centre bisects the chord.

15 min·Individual

Experiential Learning

Chord Length vs. Distance Measurement

In pairs, students use a compass to draw a large circle. They then draw several parallel chords and use a ruler to measure the length of each chord and its perpendicular distance from the centre. They record the data in a table to observe the inverse relationship between the two measurements.

25 min·Pairs

Experiential Learning

Finding the Centre of a Broken Plate

Give students a paper cutout representing a piece of a broken circular plate (an arc). They must draw any two non-parallel chords and construct their perpendicular bisectors. The point where these bisectors intersect is the centre of the original circle, reinforcing the key theorems.

20 min·Small Groups

Real-World Connections

• Designing wheels for vehicles, where spokes can be seen as radii and the axle is at the centre, ensuring balanced rotation.
• Architecture and construction, for creating arches and circular windows where understanding chords helps in ensuring structural stability and symmetry.
• Cutting circular food items like pizzas or cakes. To get equal slices, one must cut along diameters, which are the longest chords passing through the centre.
• In astronomy, calculating the path or orbit of satellites and planets, which often involves principles of circular geometry.
• Engineering design of circular components like gears, tunnels, and pipes, where placing supports and measuring clearances relies on the properties of chords.

Assessment Ideas

Exit Ticket

An 'exit ticket' where students have to solve one problem: given the radius and the length of a chord, find its distance from the centre.

Quick Check

Ask students to draw a diagram and explain in their own words why equal chords must be equidistant from the centre, without a formal proof.

Quick Check

A short quiz with a mix of direct proof questions, numerical problems based on Pythagoras theorem, and multiple-choice questions testing conceptual understanding.

Quick Check

Provide a checklist of the main theorems and concepts. Students rate their confidence level (e.g., 'I can prove it', 'I can use it', 'I need help') for each.

Frequently Asked Questions

Why do we always measure the perpendicular distance from the centre to the chord?

In geometry, the distance between a point and a line is always defined as the length of the perpendicular segment from the point to the line. This is because it is the shortest possible distance.

How is the Pythagoras theorem used in problems related to chords?

When you draw a perpendicular from the centre to a chord, it forms a right-angled triangle. The radius to one endpoint of the chord becomes the hypotenuse, the perpendicular distance is one side, and half the length of the chord is the other side. This allows you to calculate any one of these three lengths if you know the other two.

Can two chords bisect each other?

Yes, but only if they are both diameters of the circle. If two non-diameter chords bisect each other, their intersection point must be the centre of the circle, which would make them diameters.