CBSE Class 9 Maths Chapter 11 Constructions Notes

Key Points for Constructions

• Constructions exclusively use a compass and an ungraduated ruler (straightedge).
• Accuracy is paramount; even small errors can lead to incorrect results in drawings.
• Basic constructions (like bisectors and specific angles) are foundational for complex figures.
• Understanding the geometric reasoning behind each construction step helps in better retention.
• Always draw a rough sketch before starting the actual construction to plan steps and visualize the outcome.
• Use a well-sharpened pencil for fine, clear lines and arcs to ensure precision.
• Do not erase construction lines; they are essential evidence of your step-by-step working.

Key Definitions

Compass

A geometric instrument used for drawing circles or arcs and for transferring measurements.

Ruler (Straightedge)

A tool used for drawing straight lines; in constructions, its markings are typically ignored for measurement.

Angle Bisector

A ray that originates from the vertex of an angle and divides the angle into two equal angles.

Perpendicular Bisector

A line that divides a line segment into two equal parts and is also perpendicular (at 90°) to it.

Arc

A curved part of the circumference of a circle or any other curve.

Line Segment

A part of a line that has two distinct endpoints and a measurable length.

Ray

A part of a line that has one endpoint and extends infinitely in one direction.

Fundamental Geometric Constructions

Understanding the fundamental geometric constructions is the cornerstone of this chapter. These basic constructions, performed using only a compass and an ungraduated ruler, are frequently asked directly or serve as essential steps for more complex problems, especially in triangle constructions.

1. Constructing the Perpendicular Bisector of a Line Segment:

To bisect a line segment AB, first open your compass to a radius greater than half the length of AB. With A as the center, draw arcs both above and below the line segment. Repeat this step with B as the center, using the same radius, ensuring the new arcs intersect the previous ones at two distinct points, say P and Q. Join P and Q with a straight line using your ruler. This line PQ is the perpendicular bisector of AB, meaning it divides AB into two equal halves and forms a 90-degree angle with AB.

1. Constructing the Bisector of an Angle:

Given an angle $\angle ABC$, place the compass pointer at B (the vertex) and draw an arc that intersects rays BA and BC at points D and E respectively. Now, with D as the center and a suitable radius (more than half of DE), draw an arc inside the angle. With E as the center and the same radius, draw another arc that intersects the previous one at point F. Draw a ray from B passing through F. This ray BF is the angle bisector of $\angle ABC$, dividing it into two equal angles, $\angle ABF = \angle CBF$.

1. Constructing Specific Angles (e.g., 60°, 90°):

• 60° Angle: Draw a ray OA. With O as the center and any convenient radius, draw an arc intersecting OA at P. With P as the center and the same radius, draw an arc intersecting the first arc at Q. Join OQ. $\angle QOA$ is 60°.
• 90° Angle: Draw a ray OA. With O as center, draw an arc intersecting OA at P. With P as center and same radius, draw an arc intersecting the first arc at Q (which marks 60°). With Q as center and same radius, draw an arc intersecting the first arc at R (which marks 120°). Now, with Q and R as centers and a radius greater than half of QR, draw two arcs intersecting each other at S. Join OS. $\angle SOA$ is 90°. This construction uses the concept of bisecting the angle formed between the 60° and 120° marks on the initial arc.

Mastering these basics is essential before moving to more intricate triangle constructions.

Advanced Triangle Constructions

1. — Example: Construct $\triangle ABC$ where BC is the base, $\angle B$ is given, and AB + AC is given.
2. — Example: Construct $\triangle ABC$ where BC is the base, $\angle B$ is given, and AB - AC or AC - AB is given.
3. — Example: Construct $\triangle ABC$ where $\angle B$, $\angle C$, and AB + BC + CA (perimeter) are given.

Section 5

Precision is Key! In construction questions, examiners strictly look for accuracy. Use a well-sharpened pencil for thin, clear lines and arcs. Ensure your compass is tight so the radius doesn't change unintentionally. Do not erase your construction lines; they are part of your solution and show the steps taken. Clearly label all points, lines, and angles. Always start by drawing a rough sketch to plan your construction steps, especially for triangle constructions, as this helps in visualizing the final figure and avoiding errors. Practice regularly to improve speed and accuracy, and remember to justify your steps with correct geometric principles.

Quick Revision Check

• Q: What two instruments are exclusively used for classical geometric constructions? A: An ungraduated ruler (straightedge) and a compass.
• Q: What is the first crucial step when constructing a perpendicular bisector of a line segment AB? A: Open the compass to a radius greater than half the length of AB, and with A as the center, draw arcs both above and below AB.
• Q: If you need to construct a 45° angle accurately using only a compass and ruler, which basic angle constructions would you initially perform? A: First, construct a 90° angle, and then bisect it.
• Q: Why is it advised not to erase construction lines in your final drawing during an exam? A: Construction lines serve as proof of your method and the steps taken to arrive at the final figure, demonstrating your understanding of the process to the examiner.

Frequently Asked Questions