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Practical Geometry - Constructing a Copy of a Line Segment

Grade 6CBSE

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

🔑Concepts

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A line segment is a fixed portion of a line with two distinct endpoints. In geometry, a segment with endpoints AA and BB is denoted as AB‾\overline{AB}. Visually, it looks like a straight line of finite length starting at AA and ending at BB.

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To construct an accurate copy of a line segment, we primarily use a ruler and a compass. The ruler is used to draw straight lines, while the compass is used to measure and transfer distances accurately between points.

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Using a compass for copying is more precise than using a ruler to measure and redraw. A ruler might lead to errors due to the thickness of markings or parallax error, whereas a compass 'locks' the exact distance between its metal pointer and the pencil tip.

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To measure the original segment AB‾\overline{AB}, place the metal pointer of the compass on point AA and adjust the compass opening so the pencil lead exactly touches point BB. Visually, the compass legs now represent the fixed distance of the segment.

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The first step in copying is to draw a supporting line ll (which is longer than the original segment) and mark a point PP on it. This point PP will serve as the starting point of the new copy.

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Without changing the compass setting, place the pointer on PP and draw a small arc that intersects line ll at a point QQ. Visually, this creates a new segment PQ‾\overline{PQ} on line ll that is the same 'width' as the original.

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Two line segments are called congruent if they have the same length. After construction, we can state that AB‾≅PQ‾\overline{AB} \cong \overline{PQ}, which implies Length(AB)=Length(PQ)Length(AB) = Length(PQ).

📐Formulae

Length(AB‾)=Length(PQ‾)Length(\overline{AB}) = Length(\overline{PQ})

AB‾≅PQ‾\overline{AB} \cong \overline{PQ}

L=∣x2−x1∣L = |x_2 - x_1|

💡Examples

Problem 1:

Given a line segment XY‾\overline{XY} of length 6.46.4 cm, construct a copy MN‾\overline{MN} using only a ruler and a compass.

Solution:

  1. Draw a line ll and mark a point MM on it.
  2. Place the compass pointer on point XX of the given segment XY‾\overline{XY} and open it until the pencil tip reaches point YY.
  3. Maintaining the same compass width, place the metal pointer on point MM on line ll.
  4. Draw an arc that cuts the line ll at a point, and name it NN.
  5. The segment MN‾\overline{MN} is the required copy of XY‾\overline{XY}, such that Length(MN)=6.4Length(MN) = 6.4 cm.

Explanation:

The compass acts as a physical measurement transfer tool, ensuring that the distance between MM and NN is identical to the distance between XX and YY without needing to read numerical values on a ruler twice.

Problem 2:

If a line segment AB‾\overline{AB} is given, how can you construct a segment PX‾\overline{PX} whose length is twice that of AB‾\overline{AB}?

Solution:

  1. Draw a long line mm and mark a point PP on it.
  2. Measure the length of the given segment AB‾\overline{AB} using a compass (pointer on AA, pencil on BB).
  3. Place the compass pointer on PP and mark an arc on line mm to get point QQ. Now PQ=ABPQ = AB.
  4. Without changing the compass width, move the pointer to point QQ and mark another arc further along the line mm to get point XX.
  5. The segment PX‾\overline{PX} is the required segment where PX=PQ+QX=AB+AB=2×ABPX = PQ + QX = AB + AB = 2 \times AB.

Explanation:

This construction uses the addition of segments property. By placing two copies of the same segment end-to-end on a straight line, the total length becomes 2×Length(AB)2 \times Length(AB).