Practical Geometry Class 7 CBSE Mathematics

Overview: Practical Geometry

Practical Geometry introduces students to the construction of basic geometric figures using a ruler, compass, and protractor. This chapter builds on the understanding of lines, angles, and shapes learned earlier and focuses on hands-on methods to draw them accurately. It emphasizes precision and the use of geometric tools.

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Key Concepts Covered

1. Construction of a Line Parallel to a Given Line through a Point Not on the Line
2. Construction of Triangles
   • Using SSS (Side-Side-Side) criterion
   • Using SAS (Side-Angle-Side) criterion
   • Using ASA (Angle-Side-Angle) criterion
   • Using RHS (Right angle-Hypotenuse-Side) criterion for right-angled triangles

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Tools Required

• Ruler: To measure and draw straight lines of given lengths.
• Compass: To draw arcs and measure equal distances.
• Protractor: To measure and construct angles.
• Pencil: For precise markings and drawing.

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1. Construction of a Line Parallel to a Given Line through a Point Not on the Line

Objective

To draw a line parallel to a given line passing through a point that is not on the given line.

Steps

1. Draw the Given Line: Use a ruler to draw a straight line, say AB.
2. Mark the Point: Mark a point P outside the line AB (not on it).
3. Choose a Point on the Line: Mark a point C on line AB.
4. Draw an Arc from C: Using a compass, place the pointer at C and draw an arc intersecting AB at point D and extending to intersect the space near P.
5. Measure Distance CD: Keep the compass width fixed (equal to CD).
6. Draw an Arc from P: Place the compass pointer at P and draw an arc with the same radius (CD) to intersect the previous arc at point Q.
7. Join P and Q: Use a ruler to draw a straight line through P and Q. This line PQ is parallel to AB.

Key Property

• The construction uses the property that alternate angles or corresponding angles formed by a transversal with parallel lines are equal.

Example

• Given line AB and point P above it, construct PQ || AB.
• Follow the steps to get PQ parallel to AB.

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2. Construction of Triangles :Practical Geometry

A triangle can be constructed if sufficient measurements (sides and angles) are provided. This chapter covers four specific cases.

A. Construction of a Triangle Using SSS Criterion

SSS Criterion: A triangle is uniquely determined if the lengths of all three sides are given.

Steps

1. Draw the Base: Use a ruler to draw a line segment, say AB = 5 cm.
2. Construct the Second Side: Set the compass to the length of the second side, say AC = 4 cm. Place the pointer at A and draw an arc above AB.
3. Construct the Third Side: Set the compass to the length of the third side, say BC = 6 cm. Place the pointer at B and draw an arc to intersect the previous arc at point C.
4. Join the Points: Use a ruler to join A to C and B to C to form triangle ABC.

Example ON Practical Geometry

• Construct a triangle with sides 5 cm, 4 cm, and 6 cm.
• Result: Triangle ABC with AB = 5 cm, AC = 4 cm, and BC = 6 cm.

Note

• The sum of any two sides must be greater than the third side (Triangle Inequality Theorem). For 5 cm, 4 cm, and 6 cm:
  • 5 + 4 > 6 (9 > 6)
  • 5 + 6 > 4 (11 > 4)
  • 4 + 6 > 5 (10 > 5)
    This confirms the triangle is possible.

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B. Construction of a Triangle Using SAS Criterion

SAS Criterion: A triangle is uniquely determined if two sides and the angle between them are given.

Steps

1. Draw the Base: Draw a line segment, say AB = 5 cm.
2. Construct the Angle: At point A, use a protractor to measure and mark the given angle, say ∠A = 60°. Draw a ray AX from A along this angle.
3. Mark the Second Side: Set the compass to the length of the second side, say AC = 4 cm. Place the pointer at A and draw an arc on ray AX to mark point C.
4. Join the Points: Use a ruler to join B to C to complete triangle ABC.

Example

• Construct a triangle with AB = 5 cm, AC = 4 cm, and ∠A = 60°.
• Result: Triangle ABC with the specified measurements.

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C. Construction of a Triangle Using ASA Criterion

ASA Criterion: A triangle is uniquely determined if two angles and the side between them are given.

Steps

1. Draw the Base: Draw a line segment, say AB = 5 cm.
2. Construct the First Angle: At point A, use a protractor to measure and mark the first angle, say ∠A = 40°. Draw a ray AX.
3. Construct the Second Angle: At point B, measure and mark the second angle, say ∠B = 60°. Draw a ray BY.
4. Locate the Third Vertex: The rays AX and BY intersect at point C. Join A to C and B to C to form triangle ABC.

Example

• Construct a triangle with AB = 5 cm, ∠A = 40°, and ∠B = 60°.
• Result: Triangle ABC with the third angle ∠C = 180° – (40° + 60°) = 80° (by the angle sum property).

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D. Construction of a Right-Angled Triangle Using RHS Criterion

RHS Criterion: A right-angled triangle is uniquely determined if the hypotenuse and one side are given.

Steps

1. Draw the Base: Draw a line segment, say BC = 3 cm.
2. Construct the Right Angle: At point B, use a protractor to draw a perpendicular line (90°) to BC. Extend it as ray BX.
3. Mark the Hypotenuse: Set the compass to the length of the hypotenuse, say AC = 5 cm. Place the pointer at A and draw an arc to intersect ray BX at point A.
4. Join the Points: Join A to C to complete the right-angled triangle ABC.

Example

• Construct a triangle with BC = 3 cm, hypotenuse AC = 5 cm, and ∠B = 90°.
• Result: Triangle ABC, where AB = √(AC² – BC²) = √(5² – 3²) = √(25 – 9) = 4 cm (by Pythagoras theorem).

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Important Points to Remember : Practical Geometry

1. Precision: Always use sharp pencils and accurate tools for neat constructions.
2. Triangle Inequality: For any triangle, the sum of two sides must exceed the third side.
3. Angle Sum Property: The sum of angles in a triangle is always 180°.
4. Right-Angled Triangles: Verify using Pythagoras theorem (Hypotenuse² = Base² + Perpendicular²).

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Practice Questions

1. Construct a line PQ parallel to a given line XY through a point R not on XY.
2. Construct a triangle with sides 6 cm, 7 cm, and 8 cm (SSS).
3. Construct a triangle with AB = 5 cm, ∠A = 50°, and AC = 6 cm (SAS).
4. Construct a triangle with BC = 4 cm, ∠B = 45°, and ∠C = 65° (ASA).
5. Construct a right-angled triangle with hypotenuse 10 cm and one side 6 cm (RHS).

Chapter 10 Practical Geometry

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