CBSE Class 7 Maths Chapter Notes: Practical Geometry

Key Concepts & Must Remember for Practical Geometry

• Basic Construction Tools: A ruler (for drawing line segments and measuring lengths), a compass (for drawing arcs and circles), a protractor (for measuring and drawing angles), and set squares (for drawing perpendicular and parallel lines, though less common in primary constructions here).
• Line Parallel to a Given Line: A line parallel to a given line through a point not on it can be constructed using the property of equal alternate interior angles or corresponding angles.
• Triangle Construction Criteria: A unique triangle can be constructed if specific sets of measurements are given. These are SSS, SAS, ASA, and RHS.
• SSS (Side-Side-Side) Criterion: If the lengths of all three sides of a triangle are known, a unique triangle can be constructed.
• SAS (Side-Angle-Side) Criterion: If the lengths of two sides and the measure of the included angle are known, a unique triangle can be constructed.
• ASA (Angle-Side-Angle) Criterion: If two angles and the included side of a triangle are known, a unique triangle can be constructed.
• RHS (Right Angle-Hypotenuse-Side) Criterion: For a right-angled triangle, if the hypotenuse and one side are known, a unique triangle can be constructed.
• Accuracy and Neatness: Precision in drawing and measuring is paramount in practical geometry. Use a sharp pencil and ensure clean, labeled diagrams.

Essential Geometric Terms

Line Segment

A part of a line that is bounded by two distinct end points, having a definite length.

Ray

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

Parallel Lines

Two lines in a plane that never meet or intersect, no matter how far they are extended. They maintain a constant distance from each other.

Perpendicular Lines

Two lines that intersect to form a right angle (90 degrees) at their point of intersection.

Included Angle

In a triangle, the angle formed by two given sides. For example, in SAS criterion, the angle must be between the two given sides.

Included Side

In a triangle, the side connecting two given angles. For example, in ASA criterion, the side must be between the two given angles.

Hypotenuse

The longest side of a right-angled triangle, opposite to the right angle.

Understanding Triangle Construction Criteria

Constructing a triangle accurately requires specific information. You cannot just draw any three lines and form a triangle; there are unique conditions that guarantee a single, distinct triangle. These conditions are derived from the congruence criteria you studied earlier.

The SSS (Side-Side-Side) criterion states that if you know the lengths of all three sides (say, 5 cm, 6 cm, 7 cm), there's only one way to connect these lengths to form a triangle. You start by drawing one side, then use a compass to mark arcs for the other two sides from the endpoints of the first, where the arcs intersect forms the third vertex.

Similarly, the SAS (Side-Angle-Side) criterion allows construction when two sides and the included angle (the angle between those two sides) are known. For instance, if you have sides of 4 cm and 5 cm with a 60° angle between them, a unique triangle is formed. You draw one side, then draw the angle at one endpoint, and measure the second side along the arm of the angle.

The ASA (Angle-Side-Angle) criterion is used when two angles and the included side (the side between those two angles) are known. For example, if you have angles of 50° and 70° with a 6 cm side connecting their vertices, a unique triangle can be drawn. You draw the side, then construct the angles at each end.

Finally, for right-angled triangles, the RHS (Right angle-Hypotenuse-Side) criterion is powerful. If you know the length of the hypotenuse and one other side, along with the right angle (which is always 90°), you can uniquely construct the triangle. This is particularly useful as the right angle is a fixed value, simplifying construction. Understanding why these criteria work is key to mastering practical geometry and confidently constructing any required figure.

Steps to Construct a Line Parallel to a Given Line

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Worked Examples of Triangle Construction

• {"title":"Example 1: Constructing a Triangle (SSS Criterion)","description":"Construct ΔABC where AB = 5 cm, BC = 6 cm, and AC = 7 cm.","solutionMarkdown":"1. Draw a line segment BC of length 6 cm. \n2. With B as centre, draw an arc of radius 5 cm (for AB). \n3. With C as centre, draw an arc of radius 7 cm (for AC), intersecting the previous arc at point A. \n4. Join A to B and A to C. ΔABC is the required triangle."}
• {"title":"Example 2: Constructing a Right-Angled Triangle (RHS Criterion)","description":"Construct a right-angled triangle PQR where ∠Q = 90°, QR = 3 cm and PR = 5 cm.","solutionMarkdown":"1. Draw a line segment QR of length 3 cm. \n2. At point Q, draw a ray QX perpendicular to QR (forming a 90° angle). \n3. With R as centre, draw an arc of radius 5 cm (for PR), intersecting the ray QX at point P. \n4. Join P to R. ΔPQR is the required right-angled triangle."}

Exam Tip: Accuracy and Presentation are Key!

In practical geometry questions, simply understanding the steps is not enough; execution matters. Your diagrams must be neat, accurate, and clearly labeled. Use a sharp pencil and a good quality ruler and compass. Do not erase construction arcs completely, as they show your working. Always list the steps of construction clearly alongside your diagram, as marks are often awarded for both. If given measurements are in cm, make sure your final labels are also in cm. A slight inaccuracy in drawing angles or lengths can lead to loss of marks, so practice precise drawing. Always double-check your construction against the given criteria.

Practice Questions with Solutions

• Q: What are the four criteria for constructing a unique triangle? A: The four criteria are SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and RHS (Right angle-Hypotenuse-Side).
• Q: Why can't we construct a unique triangle if only three angles are given? A: If only three angles are given, infinitely many similar triangles (triangles with the same shape but different sizes) can be drawn. A unique triangle requires at least one side length.
• Q: What is the main property used to construct a line parallel to a given line through an external point? A: The main property used is the formation of equal alternate interior angles or equal corresponding angles when a transversal intersects two parallel lines.
• Q: Which tools are essential for geometric constructions in this chapter? A: A ruler (straightedge), a compass, and sometimes a protractor are essential tools for geometric constructions in this chapter.

Frequently Asked Questions