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Playing with Constructions - Artwork

Grade 6CBSE

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

🔑Concepts

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Geometric Tools: Use a ruler (straightedge) for line segments, a compass for drawing arcs and circles, and a protractor for measuring angles. Set-squares are used to draw parallel and perpendicular lines.

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Circle Construction: To draw a circle of radius rr, fix the compass needle at a center point OO and rotate the pencil point. The distance from the center to the boundary is the radius rr.

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Line Segment: A part of a line with two endpoints. To construct a segment of length ll, use a ruler to mark the start and end points.

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Perpendicular Bisector: A line that divides a given line segment into two equal halves and meets it at 90∘90^\circ. To construct it, draw arcs of radius r>12×length of segmentr > \frac{1}{2} \times \text{length of segment} from both endpoints.

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Angle Construction: Specific angles like 60∘60^\circ, 120∘120^\circ, and 90∘90^\circ can be constructed using only a compass and ruler. Bisecting a 60∘60^\circ angle gives 30∘30^\circ.

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Artistic Patterns: By combining arcs and circles of the same radius, we can create symmetrical patterns. For example, drawing six arcs of the same radius rr around the circumference of a circle of radius rr creates a perfect six-petaled flower design.

📐Formulae

d=2×rd = 2 \times r

Measure of Bisected Angle=Original Angle2\text{Measure of Bisected Angle} = \frac{\text{Original Angle}}{2}

Radius of arc for bisector>12×Length of Segment\text{Radius of arc for bisector} > \frac{1}{2} \times \text{Length of Segment}

💡Examples

Problem 1:

Construct a circle with a radius of 4 cm4 \text{ cm} and create a design by drawing another circle of the same radius with its center on the circumference of the first circle.

Solution:

  1. Use a ruler to set the compass width to 4 cm4 \text{ cm}.
  2. Draw a circle with center OO.
  3. Pick any point PP on the circumference and, without changing the compass width, draw another circle with center PP.

Explanation:

This creates an interlocking circle pattern. The distance between the two centers OO and PP is exactly r=4 cmr = 4 \text{ cm}.

Problem 2:

If a line segment XYXY measures 8 cm8 \text{ cm}, find the length of each part if it is divided by its perpendicular bisector.

Solution:

Length of each part=8 cm24 cm\begin{array}{r} \text{Length of each part} = \frac{8 \text{ cm}}{2} \\ \hline 4 \text{ cm} \end{array}

Explanation:

A perpendicular bisector divides a line segment into two equal lengths.

Problem 3:

Construct a 90∘90^\circ angle and then bisect it to create a 45∘45^\circ angle for an artwork corner.

Solution:

  1. Draw a line and construct a 90∘90^\circ angle using a compass (bisecting the angle between 60∘60^\circ and 120∘120^\circ arcs).
  2. Place the compass needle at the intersection of the 90∘90^\circ ray and the initial arc.
  3. Draw an arc.
  4. Place the needle at the start of the initial arc and draw another arc intersecting the previous one.
  5. Connect the vertex to this intersection point.

Explanation:

Bisecting a right angle (90∘90^\circ) always results in two 45∘45^\circ angles, which are useful for creating square-based geometric art.