Introduction
A parallelogram is a quadrilateral with both pairs of opposite sides parallel and equal. Special types of parallelograms include squares, rectangles, and rhombuses.
In this lesson, we will learn how to construct various parallelograms under different given conditions using geometric instruments like rulers, protractors, and compasses.
Properties of Parallelograms:
- Opposite sides are equal and parallel
- Opposite angles are equal
- Consecutive angles are supplementary
- Diagonals bisect each other
Special Parallelograms:
- Square: All sides equal, all angles 90°
- Rectangle: Opposite sides equal, all angles 90°
- Rhombus: All sides equal, opposite angles equal
Square with Given Side
Constructing a Square with Given Side Length
Steps to Construct Square ABCD with AB = 6.5cm:
- Draw side AB = 6.5cm using a ruler
- At point A, construct a 90° angle using a protractor or compass
- From point A, draw AD = 6.5cm along the constructed angle
- At point B, construct another 90° angle
- From point B, draw BC = 6.5cm along this new angle
- Join points C and D to complete the square
- Measure the diagonal AC or BD and record its length
Example: Square ABCD with AB = 6.5cm
After construction, the diagonal should measure approximately:
\[ AC = BD = 6.5\sqrt{2} \approx 9.19 \text{cm} \](Note: You'll provide the actual construction diagram image later)
Square with Given Diagonal
Constructing a Square with Given Diagonal
Steps to Construct Square with Diagonal Length d:
- Draw the diagonal AC of given length (e.g., 8cm)
- Find the midpoint O of diagonal AC using perpendicular bisector
- Construct a perpendicular bisector through point O
- With O as center and radius OA, draw a circle intersecting the bisector at B and D
- Join points A to B, B to C, C to D, and D to A to complete the square
Key Properties:
The diagonals of a square:
- Are equal in length
- Bisect each other at 90°
- Bisect the angles of the square
Rectangle with Given Side
Constructing a Rectangle with Given Sides
Steps to Construct Rectangle with Length l and Width w:
- Draw side AB = length (l) using a ruler
- At point A, construct a 90° angle
- From point A, draw AD = width (w) along the constructed angle
- At point B, construct another 90° angle
- From point B, draw BC = width (w) along this new angle
- Join points C and D to complete the rectangle
Example: Rectangle with l = 7cm, w = 4cm
The constructed rectangle should have:
- Opposite sides equal (AB = CD = 7cm, AD = BC = 4cm)
- All four angles equal to 90°
- Diagonals AC and BD equal in length
Rectangle with Side and Diagonal
Constructing a Rectangle with Given Side and Diagonal
Steps to Construct Rectangle with Side AB and Diagonal AC:
- Draw side AB of given length
- At point A, construct a 90° angle
- With point A as center and diagonal length as radius, draw an arc
- From point B, draw a perpendicular to meet the arc at point C
- From point C, draw CD parallel and equal to AB
- Join point D to A to complete the rectangle
Verification:
After construction, verify that:
- Opposite sides are equal and parallel
- All angles are 90°
- The measured diagonal matches the given length
Parallelogram with Sides and Angle
Constructing a Parallelogram with Given Sides and Angle
Steps to Construct Parallelogram with Sides a, b and Included Angle θ:
- Draw side AB = length a
- At point A, construct angle θ using a protractor
- From point A, draw AD = length b along the constructed angle
- With point B as center and radius b, draw an arc
- With point D as center and radius a, draw another arc intersecting the first arc at C
- Join points B to C and C to D to complete the parallelogram
Example: Parallelogram with a = 5cm, b = 3cm, θ = 60°
The constructed parallelogram should have:
- AB = CD = 5cm
- AD = BC = 3cm
- ∠A = ∠C = 60°
- ∠B = ∠D = 120° (consecutive angles supplementary)
Regular Compound Plane Shapes
Constructing Regular Compound Plane Shapes
Steps to Construct Compound Shapes:
- Analyze the compound shape to identify basic components (squares, rectangles, etc.)
- Construct the base shape first using appropriate methods
- Add adjacent shapes by sharing sides or angles as specified
- Ensure all given dimensions and angles are accurately represented
- Verify all sides and angles meet the given conditions
Example Compound Shapes:
- Two squares sharing a common side
- Rectangle with an equilateral triangle on one side
- Combination of parallelograms forming a larger shape
Practice Exercise
Question 1
Construct a square PQRS with PQ = 5cm. Measure and record the length of its diagonal PR.
Steps:
- Draw PQ = 5cm
- At P, construct 90° and draw PS = 5cm
- At Q, construct 90° and draw QR = 5cm
- Join R to S
- Measure diagonal PR ≈ 7.07cm
Question 2
Construct a square with diagonal 10cm. Measure and record the length of its sides.
Steps:
- Draw diagonal AC = 10cm
- Find midpoint O
- Construct perpendicular bisector through O
- With radius OA, draw circle intersecting bisector at B and D
- Join points to form square ABCD
- Measure sides ≈ 7.07cm each
Question 3
Construct rectangle ABCD with AB = 8cm and BC = 5cm. Measure its diagonals.
Steps:
- Draw AB = 8cm
- At B, construct 90° and draw BC = 5cm
- At A, construct 90° and draw AD = 5cm
- Join D to C
- Measure diagonals ≈ 9.43cm each
Question 4
Construct a parallelogram EFGH with EF = 6cm, FG = 4cm, and ∠EFG = 75°.
Steps:
- Draw EF = 6cm
- At F, construct 75° and draw FG = 4cm
- With E as center, radius 4cm draw arc
- With G as center, radius 6cm draw arc intersecting at H
- Join H to E and G
Question 5
Construct a compound shape consisting of a square (4cm side) adjacent to a rectangle (4cm × 6cm) sharing one full side.
Steps:
- Construct square ABCD with AB = 4cm
- Extend side BC to E where CE = 2cm (total BE = 6cm)
- At E, construct 90° and draw EF = 4cm
- Join F to D
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