NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions
Question 76

Question. 76

Draw a line segment of length 6.5 cm and divide it into four equal parts, using ruler and compasses.

Answer:

Construction: (1) Draw AB = 6.5 cm. (2) Construct its perpendicular bisector to get midpoint M → AM = MB = 3.25 cm. (3) Bisect AM and MB again to obtain points that divide AB into four equal parts of 1.625 cm each.

Detailed Answer with Explanation:

Q76. Draw a line segment of length 6.5 cm and divide it into four equal parts using a ruler and compasses.

Simple Tools Needed

  • Ruler (with centimetres)
  • Compass
  • Pencil

Construction (Step by Step)

Step 1: Draw the segment.

Use the ruler to draw a straight line segment AB such that

( AB = 6.5 ext{ cm} )

Step 2: Find the midpoint of AB (bisect AB).

  1. Put the compass point on A. Open it a little more than half of AB. Draw arcs above and below the line.
  2. Without changing the compass width, put the compass point on B. Draw arcs to cross the previous arcs.
  3. Join the intersection points of the arcs with a straight line. It cuts AB at M. This line is the perpendicular bisector.

Now M is the midpoint, so:

( AM = MB )

( AM = MB = dfrac{6.5}{2} ext{ cm} )

( AM = MB = 3.25 ext{ cm} )

Step 3: Bisect AM to get the first quarter point.

  1. Repeat the same “arc—arc—join” method on the smaller segment AM.
  2. The midpoint of AM is a new point, say P, on AB.

Then:

( AP = PM )

( AP = PM = dfrac{AM}{2} )

( AP = PM = dfrac{3.25}{2} ext{ cm} )

( AP = PM = 1.625 ext{ cm} )

Step 4: Bisect MB to get the second quarter point.

  1. Repeat the same bisection method on the segment MB.
  2. The midpoint of MB is another new point, say Q, on AB.

Then:

( MQ = QB )

( MQ = QB = dfrac{MB}{2} )

( MQ = QB = dfrac{3.25}{2} ext{ cm} )

( MQ = QB = 1.625 ext{ cm} )

Step 5: Read the four equal parts on AB.

The points on AB are now in order: A, P, M, Q, B.

So the four equal pieces are:

( AP,; PM,; MQ,; QB )

( AP = PM = MQ = QB = 1.625 ext{ cm} )

Why this works (Very simple idea)

First, we split the whole length into two equal halves (bisection). Then we split each half into two equal halves again. Halving twice gives four equal parts.

( ext{Quarter} = dfrac{1}{2} imesdfrac{1}{2} = dfrac{1}{4} )

( dfrac{1}{4} imes 6.5 = 1.625 ext{ cm} )

Verification (Optional but helpful)

Measure each part with the ruler. You should see:

( AP approx 1.625 ext{ cm},; PM approx 1.625 ext{ cm},; MQ approx 1.625 ext{ cm},; QB approx 1.625 ext{ cm} )

Adding all four should give back the original length:

( 1.625 + 1.625 + 1.625 + 1.625 = 6.5 ext{ cm} )

Result: The line segment of (6.5 ext{ cm}) is divided into four equal parts of (1.625 ext{ cm}) each using only a ruler and compasses.
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NCERT Exemplar Solutions Class 6 – Mathematics – Unit 9: Symmetry and Practical Geometry – Problems and Solutions | Detailed Answers