Class 8 Math Practical Geometry Notes

sBagul

2 days ago

Practical Geometry – Class 8

Hi everyone! This chapter is all about Practical Geometry. We’ll learn how to construct different quadrilaterals (four-sided figures) using a ruler, compass, and protractor.

What is Construction?

Geometric construction involves drawing shapes and figures using precise tools and following specific steps. We’ll be focusing on constructing quadrilaterals.

Constructing Quadrilaterals

To construct a unique quadrilateral, you’ll generally need five pieces of information. This information might include:

Lengths of four sides and one diagonal
Lengths of three sides and two diagonals
Lengths of two adjacent sides and three angles
Lengths of three sides and the two included angles
When special properties are given (like a square or rhombus)

Let’s look at some examples:

1. Constructing a Quadrilateral When Four Sides and One Diagonal are Given:

Construct quadrilateral ABCD where AB = 4 cm, BC = 5 cm, CD = 6 cm, DA = 5 cm and AC = 7 cm.

Draw side AB = 4 cm.
With A as center and radius 7 cm (diagonal AC), draw an arc.
With B as center and radius 5 cm (side BC), draw another arc intersecting the previous arc at C.
With C as center and radius 6 cm (side CD), draw an arc.
With A as center and radius 5 cm (side DA), draw another arc intersecting the previous arc at D.
Join AC, BC, CD, and DA.

2. Constructing a Quadrilateral When Three Sides and Two Diagonals are Given:

Construct quadrilateral PQRS where PQ = 4cm, QR = 6cm, RS = 5cm, PR = 7cm and QS = 8cm.

Draw side PQ = 4cm.
With P as center and radius 7cm (diagonal PR), draw an arc.
With Q as center and radius 8cm (diagonal QS), draw another arc intersecting the previous arc at R.
With Q as center and radius 6cm (side QR), draw an arc.
With R as center and radius 5cm (side RS), draw another arc intersecting the previous arc at S.
Join PR, QR, QS, RS, and SP.

(Diagram would be here)

3. Constructing a Parallelogram when Two Adjacent Sides and One Angle are given.

Construct parallelogram ABCD where AB = 5cm, BC = 4cm and ∠B = 60°.

Draw AB = 5cm.
At B, construct ∠B = 60°.
Along the ray of ∠B, mark BC = 4cm.
With A as center and radius 4cm, draw an arc.
With C as center and radius 5cm, draw an arc intersecting the previous arc at D.
Join AD and CD.

(Diagram would be here)

Tips for Construction

Use sharp pencils for accurate drawings.
Measure lengths and angles carefully.
Follow the steps in the correct order.
Practice makes perfect!

Applications of Practical Geometry

1. Engineering and Architecture:

Creating blueprints and technical drawings.

2. Design and Art:

Creating patterns, designing graphics.

3. Construction:

Laying out building plans.

Practical geometry is a valuable skill for many fields and helps develop spatial reasoning.

Practical Geometry Quiz – Tough Application Problems

1. Parallelogram Construction: Construct a parallelogram ABCD where AB = 6 cm, BC = 4 cm, and angle B = 75 degrees. What is the length of diagonal AC?

Approximately 7.2 cm (students should measure this after construction)

Students should follow the construction steps for a parallelogram given two sides and an included angle. The length of the diagonal should be measured with a ruler after construction.

2. Rhombus Construction: Construct a rhombus PQRS where PR = 8 cm and QS = 6 cm. What is the length of each side of the rhombus?

5 cm

The diagonals of a rhombus bisect each other at right angles. Half of PR is 4 cm, and half of QS is 3 cm. Using the Pythagorean theorem, the side of the rhombus is √(4² + 3²) = 5 cm. Students should verify this after construction.

3. Rectangle Construction: Construct a rectangle ABCD where AB = 5 cm and AC = 7 cm. What is the length of BC?

Approximately 4.9 cm (students should measure this after construction)

In a rectangle, the diagonals are equal. Use the Pythagorean theorem to find BC: BC = √(AC² – AB²) = √(7² – 5²) = √24 ≈ 4.9 cm. Students should verify this after construction.

4. Square Construction: Construct a square ABCD where the diagonal AC = 6 cm. What is the length of each side of the square?

Approximately 4.2 cm (students should measure this after construction)

In a square, the diagonal is √2 times the side. Side = Diagonal / √2 = 6 / √2 ≈ 4.2 cm. Students should verify this after construction.

5. Quadrilateral with Specific Angles: Construct quadrilateral ABCD where AB = 4 cm, BC = 5 cm, angle A = 90 degrees, and angle B = 110 degrees. What is the measure of angle C?

Students should measure angle C after construction.

The sum of angles in a quadrilateral is 360 degrees. Students should construct the quadrilateral and measure angle C with a protractor. Angle C = 360 – 90 – 110 – Angle D (which they will also measure after construction).

6. Trapezium Construction: Construct trapezium ABCD where AB is parallel to CD, AB = 8 cm, BC = 5 cm, CD = 4 cm, and AD = 6 cm. What is the height of the trapezium? (This requires additional construction)

Students should measure the height after construction.

Construct the trapezium. To find the height, draw a perpendicular from C (or D) to AB. Measure the length of this perpendicular.

7. Kite Construction: Construct kite ABCD where AB = AD = 5 cm and BC = CD = 7 cm. Measure the length of the diagonals AC and BD.

Students should measure AC and BD after construction.</div

Students should follow the steps to construct a kite and then measure its diagonals.

8. Cyclic Quadrilateral Construction: Construct a cyclic quadrilateral ABCD where AB = 4 cm, BC = 5 cm, CD = 6 cm, and AD = 3 cm. What is the sum of angles A and C?

180 degrees

In a cyclic quadrilateral, opposite angles are supplementary (add up to 180 degrees). Therefore, angle A + angle C = 180 degrees. Students should verify after construction.

9. Constructing a Quadrilateral with Given Angles and Sides: Construct a quadrilateral ABCD where AB = 4 cm, ∠A = 60°, ∠B = 90°, BC = 5 cm and ∠C = 120°. Measure the length of AD.

Students should measure AD after construction.

Follow the steps for constructing a quadrilateral given sides and angles. The length of AD should be measured with a ruler after construction.

10. Area Calculation (Post Construction): Construct a rectangle ABCD with AB = 6cm and BC = 4cm. Calculate its area and verify by measuring the dimensions after construction.

Area = 24 sq cm</div

Area of a rectangle = length * width = 6cm * 4cm = 24 sq cm. Students should construct and measure to verify.