NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions
Question 84

Question. 84

Draw an angle of 60° using ruler and compasses and divide it into four equal parts. Measure each part.

Answer:

Each part = 15°.

Detailed Answer with Explanation:

Q84. Draw 60° and divide it into four equal parts. Measure each part.

Goal

We will first draw an angle of 60°, then use angle bisection twice to split it into four equal parts.

What you need

  • Ruler (scale)
  • Compass
  • Pencil
  • Protractor (only for checking the final result)

Part A — Draw an angle of 60°

  1. Draw a base ray.
    Draw a straight line and mark a point O on it. From O, draw ray OA to the right.
  2. Make an arc with any radius.
    Place the compass point at O. Take a convenient radius (not too small). Draw an arc cutting ray OA at point P.
  3. Mark off 60° using equal chords.
    Without changing the compass width, place the compass point at P and cut the first arc at point Q.
    Join O to Q. The angle ∠QOA is 60°.

( angle QOA = 60^circ )

Part B — Bisect 60° to get 30°

  1. Draw equal arcs from the arms.
    With the same or any convenient radius, draw an arc centered at O to cut both arms OA and OQ at points P and R.
  2. Find the midpoint of the arc chord.
    With compass at P, draw a small arc inside the angle. With the same radius at R, draw another arc to intersect at M.
  3. Draw the bisector.
    Join O to M. Ray OM bisects 60° into two equal angles of 30° each.

( 60^circ div 2 = 30^circ )

( angle MOA = 30^circ,quad angle QOM = 30^circ )

Part C — Bisect each 30° again to get 15°

  1. Bisect the left 30°:
    Repeat the same bisection steps in the angle between OA and OM to draw a new ray (say OX). This makes two equal angles of 15° here.
  2. Bisect the right 30°:
    Repeat the bisection in the angle between OM and OQ to draw another ray (say OY). This also makes two equal angles of 15°.

( 30^circ div 2 = 15^circ )

( angle XOA = 15^circ,quad angle MOX = 15^circ,quad angle YO M = 15^circ,quad angle QOY = 15^circ )

Why this works (in simple words)

Each time we draw an angle bisector, it splits the angle into two equal halves. First we split 60° into two 30° angles. Then we split each 30° into two 15° angles. So we get four equal parts.

( 60^circ = 4 imes 15^circ )

Check with a protractor (optional)

Place the protractor center at O and the baseline along OA. You should see marks close to 15°, 30°, 45°, and 60° at the four rays.

Final Answer

Each part = 15°.

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NCERT Exemplar Solutions Class 6 – Mathematics – Unit 9: Symmetry and Practical Geometry – Problems and Solutions | Detailed Answers