CBSE Class 8 Maths: Applied Practical Geometry Ex 4.5 - Constructing Special Quadrilaterals

Welcome, Class 8 students, to an exciting part of your Maths journey: Applied Practical Geometry, specifically focusing on Exercise 4.5! In previous classes, you learned to construct basic shapes like triangles. Now, we're stepping up to the fascinating world of quadrilaterals. This chapter isn't just about drawing; it's about understanding the unique properties of shapes like squares, rectangles, rhombuses, and parallelograms, and then using those properties to construct them accurately. You'll discover that for these special quadrilaterals, you often need fewer measurements because their inherent characteristics fill in the gaps. By the end of this page, you'll be a pro at constructing these shapes using just a ruler, compass, and your smart geometric thinking! Let's dive in and build some fantastic figures!

Understanding Construction of Special Quadrilaterals

In practical geometry, you learn how to draw various geometric figures accurately using tools like a ruler, compass, and protractor. While constructing a general quadrilateral usually requires five independent measurements (like side lengths and diagonal lengths, or side lengths and angles), special quadrilaterals are different. They possess unique properties that simplify their construction, often allowing you to draw them with fewer given measurements. The key is to first recall and apply these specific properties.

Let's quickly review the properties of the special quadrilaterals you'll encounter in Exercise 4.5:

  • Square: All four sides are equal in length, and all four interior angles are 90 degrees. Its diagonals are equal, bisect each other, and are perpendicular to each other. If you know just one side length, you can construct a square.
  • Rectangle: Opposite sides are equal and parallel, and all four interior angles are 90 degrees. Its diagonals are equal and bisect each other. Knowing the lengths of two adjacent sides is enough to construct a rectangle.
  • Rhombus: All four sides are equal in length. Opposite angles are equal. Its diagonals bisect each other at right angles (90 degrees). If you are given the lengths of its two diagonals, or one side and one diagonal, you can construct a rhombus.
  • Parallelogram: Opposite sides are equal and parallel. Opposite angles are equal. Its diagonals bisect each other. To construct a parallelogram, you typically need two adjacent sides and one angle, or two adjacent sides and a diagonal.

Before you start any construction, always draw a neat, labelled rough sketch. This helps you plan your steps and visualize how the properties will be used.

Step-by-Step Construction of a Rhombus (Given Diagonals)

  1. Step 1: Draw one diagonal — Let's construct a Rhombus BEST where the diagonals are BE = 4.5 cm and ST = 6 cm. First, draw the diagonal ST of length 6 cm. Use your ruler to make sure it's exactly 6 cm.
  2. Step 2: Construct the perpendicular bisector — We know that the diagonals of a rhombus bisect each other at right angles. So, we need to find the midpoint of ST and draw a line perpendicular to it. To do this, open your compass to a radius more than half of ST (e.g., 4 cm). Place the compass needle on S and draw arcs above and below ST. Repeat with the compass needle on T, using the same radius. The two arcs will intersect at two points. Draw a straight line connecting these two intersection points. This line is the perpendicular bisector of ST. Mark the point where it intersects ST as O. O is the midpoint of ST.
  3. Step 3: Mark the other diagonal's endpoints — The other diagonal, BE, has a length of 4.5 cm. Since the diagonals bisect each other, half of BE will be 4.5 cm / 2 = 2.25 cm. Now, place the compass needle on point O. Open the compass to a radius of 2.25 cm. Draw an arc on the perpendicular bisector above ST, and another arc below ST. Mark the intersection points as B and E respectively. These are the endpoints of the second diagonal.
  4. Step 4: Join the vertices — Finally, connect the points B to S, S to E, E to T, and T to B with your ruler to form the rhombus BEST. You have successfully constructed a rhombus using the properties of its diagonals!

Exam Tip: Accuracy and Planning are Key!

When attempting practical geometry questions in your exams, remember that precision is paramount. Always start by drawing a rough sketch of the quadrilateral you need to construct. Label its vertices and mark the given measurements. Crucially, recall and list the properties of that specific type of quadrilateral (square, rectangle, rhombus, parallelogram) before you even pick up your tools. This planning helps you identify which property will guide your construction steps.

Use a sharp pencil, a well-calibrated ruler, and a sturdy compass and protractor. Make sure your lines are thin and clear. Avoid pressing too hard, as corrections can become messy. Do not erase your construction lines unless explicitly asked; they demonstrate your understanding of the process. Finally, after completing the construction, recheck your figure against the given measurements and the properties of the quadrilateral to ensure accuracy. A small error in measurement can lead to a completely different looking figure!

Practice Questions with Solutions

  • Q: Construct a square READ with RE = 5.1 cm. A: Step 1: Draw a line segment RE of length 5.1 cm. Step 2: At point R, construct a 90° angle. Use a protractor or compass for accuracy. Step 3: With R as the center and radius 5.1 cm, draw an arc on the 90° line to get point A. Step 4: At point E, construct a 90° angle. Step 5: With E as the center and radius 5.1 cm, draw an arc on the 90° line to get point D. Step 6: Join A and D. This completes the square READ. Verify that AD = 5.1 cm and all angles are 90°. Final answer: Square READ is constructed.
  • Q: Construct a rhombus ABCD where the diagonals are 5.2 cm and 6.4 cm long. A: Step 1: Draw one diagonal, say AC = 6.4 cm. Step 2: Construct the perpendicular bisector of AC. Mark the intersection point as O (midpoint). Step 3: Since diagonals of a rhombus bisect each other perpendicularly, the other diagonal BD (5.2 cm) will be bisected into two parts of 5.2/2 = 2.6 cm each. Step 4: With O as the center and radius 2.6 cm, draw arcs on both sides of AC along the perpendicular bisector. Mark these points as B and D. Step 5: Join A to B, B to C, C to D, and D to A to form the rhombus ABCD. Final answer: Rhombus ABCD is constructed.
  • Q: Construct a rectangle with adjacent sides of lengths 5 cm and 4 cm. A: Step 1: Draw a line segment AB of length 5 cm. Step 2: At point A, construct a 90° angle (since all angles of a rectangle are 90°). Step 3: With A as the center and radius 4 cm (the other adjacent side), draw an arc on the 90° line to mark point D. Step 4: At point B, construct a 90° angle. Step 5: With B as the center and radius 4 cm, draw an arc on the 90° line to mark point C. Step 6: Join D and C. Verify that DC = 5 cm. This completes the rectangle ABCD. Final answer: Rectangle ABCD is constructed.
  • Q: Construct a parallelogram ABCD where AB = 6 cm, BC = 4 cm, and ∠ABC = 60°. A: Step 1: Draw a line segment AB of length 6 cm. Step 2: At point B, construct an angle of 60° using a protractor or compass. Step 3: Along the ray of the 60° angle, measure 4 cm from B and mark point C. (BC = 4 cm). Step 4: We know that opposite sides of a parallelogram are equal and parallel. So, AD will be 4 cm and CD will be 6 cm. Step 5: With A as the center and radius 4 cm, draw an arc. Step 6: With C as the center and radius 6 cm, draw an arc intersecting the previous arc. Mark the intersection point as D. Step 7: Join AD and CD. This forms the parallelogram ABCD. Final answer: Parallelogram ABCD is constructed.

Frequently Asked Questions

What is the main difference between constructing a general quadrilateral and a special quadrilateral?

For a general quadrilateral, you typically need five independent measurements (like four sides and one diagonal) to construct it uniquely. However, for special quadrilaterals like squares, rectangles, or rhombuses, you need fewer measurements because their inherent geometric properties (e.g., all angles are 90 degrees in a square) provide the missing information.

Why is a rough sketch important before starting construction?

A rough sketch helps you visualize the figure and plan the steps for construction. It allows you to clearly label the given measurements and mark the properties that you will use, preventing mistakes and ensuring an organized approach to the problem.

What tools are essential for practical geometry constructions?

The essential tools for practical geometry include a sharp pencil, a straight edge (ruler), a compass, and a protractor. Using these tools accurately is crucial for producing precise and correct geometric figures.

Can I erase construction lines after completing the figure?

Generally, it's best not to erase construction lines unless specifically instructed to do so. These lines demonstrate your understanding of the construction process and show how you arrived at the final figure. Keep them light and clear.