Mathematics · Class 9 · Constructions · Term 3

Triangle Construction: Base, Base Angle, and Sum of Sides

Learn the method to construct a triangle when its base, a base angle, and the sum of the other two sides are given.

TL;DR:Unlock a fundamental secret of circles that beautifully connects an arc, the circle's centre, and any point on its boundary.

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

About This Topic

This topic, 'Angle Subtended by an Arc', is a cornerstone of circle geometry in the Class 9 curriculum, as outlined by NCERT and followed by boards like CBSE. It builds upon students' prior understanding of basic circle components like arcs, chords, and radii, and introduces a fundamental relationship between arcs and the angles they form. The central theorem states that the angle an arc subtends at the centre is precisely double the angle it subtends at any point on the remaining part of the circle. This concept is not just an isolated theorem but a gateway to understanding subsequent properties, such as 'angles in the same segment are equal' and the special case that 'the angle in a semicircle is a right angle'.

For the teacher, the pedagogical approach should shift from rote memorisation of the theorem to intuitive discovery and logical proof. By engaging students in activities that allow them to measure and compare these angles, they can first hypothesise the relationship before delving into the formal proof. The proof itself is an excellent exercise in applying properties of triangles, particularly isosceles triangles and the exterior angle theorem. Mastering this topic is crucial as it lays the groundwork for more advanced concepts in Class 10, including tangents and cyclic quadrilaterals, and enhances students' spatial reasoning and deductive logic skills.

Key Questions

1. Explain the role of the perpendicular bisector in this construction.
2. Justify why the resulting triangle meets all the given conditions.
3. Analyse the key steps of the construction and the geometric reasoning for each.

Learning Objectives

• State and explain the theorem that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
• Prove the theorem for different cases of arcs (minor, major, and semicircle).
• Apply the theorem to calculate unknown angles in geometric figures involving circles.
• Deduce and prove that the angle in a semicircle is a right angle.
• Solve problems based on the property that angles in the same segment are equal.

Key Vocabulary

Arc	A continuous piece of the circumference of a circle.
Subtend	To form an angle at a point. An arc or a line segment subtends an angle at a point if the point is the vertex of the angle and the ends of the arc/segment lie on the arms of the angle.
Central Angle	An angle whose vertex is the centre of a circle and whose sides are two radii.
Inscribed Angle	An angle formed by two chords in a circle that have a common endpoint on the circumference.
Segment of a Circle	The region enclosed by an arc and its corresponding chord.

Watch Out for These Misconceptions

Common Misconception

Students believe that the angle subtended at the circumference changes if the point is moved along the same arc.

What to Teach Instead

The angle subtended by a specific arc at any point on the remaining part of the circle is constant. This is a direct consequence of the main theorem and leads to the property that 'angles in the same segment are equal'.

Common Misconception

Confusing the angle subtended by the minor arc with the reflex angle subtended by the major arc at the centre.

What to Teach Instead

Clarify that the angle subtended by a minor arc is the interior angle at the centre (< 180°), while the angle subtended by the major arc is the corresponding reflex angle (> 180°). The inscribed angle is always related to the arc on the opposite side.

Common Misconception

Applying the theorem to angles whose vertex is not on the circumference or not at the centre.

What to Teach Instead

The theorem has very specific conditions: one vertex must be at the centre and the other must be on the remaining part of the circle's circumference. An angle formed by intersecting chords inside the circle follows a different rule.

Active Learning Ideas

Experiential Learning

Paper Plate Geometry

Students use paper plates to represent circles. They mark an arc, fold the plate to find the centre, and draw the central angle. They then pick a point on the circumference and draw the inscribed angle, measuring both with a protractor to discover the 2:1 relationship.

25 min·Pairs

Experiential Learning

Dynamic Geometry Exploration

Using software like GeoGebra, students construct a circle and an arc. They can then create an inscribed angle and drag the vertex along the circumference, observing that the angle measure remains constant and is always half the central angle.

20 min·Individual

Experiential Learning

String and Board Discovery

On a corkboard, students fix three pins to form a circle's centre and the endpoints of an arc. They use string to represent the angles at the centre and at another pin placed on the circumference, comparing the angles formed.

30 min·Small Groups

Real-World Connections

• Architectural Design: Used in designing arches, domes, and curved windows to ensure structural integrity and aesthetic appeal.
• Satellite Communication: Calculating the 'field of view' or coverage area of a satellite, where the Earth's curvature forms an arc and the satellite is a point.
• Navigation and Surveying: The principle is related to triangulation techniques used to pinpoint a location based on angles to known landmarks.
• Sports Field Design: Marking the three-point line in a basketball court or the 'D' in a hockey field involves constructing arcs that subtend specific angles.
• Photography: Understanding the angle of view of a camera lens can be modelled using a circle, where the lens is a point on the circumference and the subject is an arc.

Assessment Ideas

Quick Check

Give students a worksheet with diagrams of circles with missing angles. They must find the angles and provide the theorem as a reason. This can be done as a 'pair-check' activity.

Quick Check

A section in the unit test with problems that require direct application of the theorem, its corollaries (angle in a semicircle), and multi-step problems combining it with triangle properties.

Exit Ticket

Provide an 'exit ticket' with two problems: one straightforward application and one slightly more complex. Students solve it and then check their work against a provided solution to gauge their own understanding.

Frequently Asked Questions

Why is the angle in a semicircle always 90 degrees?

A semicircle is an arc whose endpoints form a diameter. This arc subtends a straight angle, which is 180 degrees, at the centre. According to the theorem, the angle subtended at any point on the circumference is half the central angle, so it is half of 180 degrees, which is always 90 degrees.

Does the theorem work for a major arc as well?

Yes, it does. The angle subtended by a major arc at the centre is a reflex angle (greater than 180 degrees). The angle it subtends at any point on the remaining part of the circle (which is the minor arc) will be exactly half of this reflex angle.

What if we have two different points on the circumference? Will the angles be the same?

Yes, as long as both points are on the same 'remaining part' of the circle relative to the arc. This leads to another important theorem: Angles in the same segment of a circle are equal.