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Playing with Constructions - Constructing Squares and Rectangles

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A square is a special quadrilateral where all four sides are equal and each interior angle is exactly 90∘90^{\circ}. To construct it, we need only one side length.

Properties of a square showing 90 degree angles and equal sides.
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A rectangle is a quadrilateral where opposite sides are equal and all angles are 90∘90^{\circ}. Construction requires the length (ll) and the breadth (bb).

Rectangle diagram with length and breadth labeled.
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The construction process involves using a ruler to draw the base and a protractor or compass to ensure 90∘90^{\circ} corners.

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Diagonal property: In both squares and rectangles, the diagonals bisect each other. In a square, diagonals are also perpendicular and equal.

📐Formulae

Perimeter of Square=4×s\text{Perimeter of Square} = 4 \times s

Area of Square=s2\text{Area of Square} = s^2

Perimeter of Rectangle=2×(l+b)\text{Perimeter of Rectangle} = 2 \times (l + b)

Area of Rectangle=l×b\text{Area of Rectangle} = l \times b

Interior Angle=90∘\text{Interior Angle} = 90^{\circ}

💡Examples

Problem 1:

Construct a square ABCDABCD with side length 4 cm4\text{ cm}.

Solution:

  1. Draw a line segment AB=4 cmAB = 4\text{ cm}.
  2. At point AA, construct an angle of 90∘90^{\circ} using a compass or protractor.
  3. From AA, cut an arc of radius 4 cm4\text{ cm} on the perpendicular line to find point DD.
  4. Similarly, at point BB, construct an angle of 90∘90^{\circ} and cut an arc of 4 cm4\text{ cm} to find point CC.
  5. Join CDCD. ABCDABCD is the required square.

Explanation:

Since all sides of a square are equal and all angles are 90∘90^{\circ}, we use the fixed side s=4 cms = 4\text{ cm} and right angles at the base vertices.

Problem 2:

Construct a rectangle PQRSPQRS where PQ=6 cmPQ = 6\text{ cm} and QR=4 cmQR = 4\text{ cm}.

Solution:

  1. Draw a line segment PQ=6 cmPQ = 6\text{ cm}.
  2. At QQ, draw a ray QXQX such that ∠PQX=90∘\angle PQX = 90^{\circ}.
  3. With QQ as center and radius 4 cm4\text{ cm}, draw an arc to cut QXQX at RR.
  4. From RR, draw an arc of radius 6 cm6\text{ cm} parallel to PQPQ.
  5. From PP, draw an arc of radius 4 cm4\text{ cm} to intersect the previous arc at SS.
  6. Join RSRS and PSPS.

Explanation:

In a rectangle, opposite sides are equal. Thus, PQ=RS=6 cmPQ = RS = 6\text{ cm} and QR=PS=4 cmQR = PS = 4\text{ cm}. All corners must be 90∘90^{\circ}.

Problem 3:

Construct a square WXYZWXYZ where each side measures 5 cm5\text{ cm}.

Square WXYZ with side 5 cm.

Solution:

  1. Draw a line segment WX=5 cmWX = 5\text{ cm}.
  2. At point WW, construct an angle of 90∘90^{\circ} using a protractor.
  3. Use a compass set to 5 cm5\text{ cm} to mark point ZZ on the perpendicular line.
  4. Similarly, at point XX, construct a 90∘90^{\circ} angle and mark point YY at a distance of 5 cm5\text{ cm}.
  5. Join YY and ZZ to complete the square.

Explanation:

Since all sides of a square are equal and all angles are 90∘90^{\circ}, we use the side length 5 cm5\text{ cm} for all four sides and ensure right angles at each vertex.

Problem 4:

Construct a rectangle LMNOLMNO with length LM=7 cmLM = 7\text{ cm} and width MN=3 cmMN = 3\text{ cm}.

Rectangle LMNO with length 7 cm and width 3 cm.

Solution:

  1. Draw a horizontal line segment LM=7 cmLM = 7\text{ cm}.
  2. Construct a 90∘90^{\circ} angle at point MM.
  3. From point MM, measure 3 cm3\text{ cm} upwards and mark it as point NN.
  4. Construct a 90∘90^{\circ} angle at point LL.
  5. From point LL, measure 3 cm3\text{ cm} upwards and mark it as point OO.
  6. Connect point OO to point NN.

Explanation:

A rectangle requires its opposite sides to be equal (LM=ON=7 cmLM = ON = 7\text{ cm} and MN=LO=3 cmMN = LO = 3\text{ cm}) and all angles to be 90∘90^{\circ}.