NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions
Question 82

Question. 82

Draw a circle of radius 6 cm using ruler and compasses. Draw one of its diameters. Draw the perpendicular bisector of this diameter. Does this perpendicular bisector contain another diameter of the circle?

Answer:

Yes, the perpendicular bisector of a diameter passes through the centre and forms another diameter of the circle.

Detailed Answer with Explanation:

Q82. Construction and reasoning

Question: Draw a circle of radius 6 cm using a ruler and compasses. Draw one of its diameters. Draw the perpendicular bisector of this diameter. Does this perpendicular bisector give another diameter of the circle?

What you need

  • Compass
  • Ruler (scale) with centimetres
  • Sharp pencil
  • Eraser

Step 1 — Draw the circle of radius 6 cm

  1. Open the compass to a gap of 6 cm using the ruler.
  2. Place the compass tip on a point on your paper. Mark this point as (O) (the centre).
  3. Rotate the compass once around the centre to draw the circle.

Step 2 — Draw a diameter of the circle

  1. Choose any point on the circle and mark it as (A).
  2. Place the ruler through (O) and (A). Draw a straight line all the way across the circle so it meets the circle on the other side at a point.
  3. Mark that opposite point as (B). Now (AB) is a diameter because it passes through the centre.

Step 3 — Draw the perpendicular bisector of the diameter (AB)

  1. Keep the compass opening a little more than half of (AB).
  2. With the compass tip on (A), draw two small arcs above and below the line (AB).
  3. Without changing the compass width, put the compass tip on (B) and draw two arcs that cross the previous arcs.
  4. Mark the arc intersection above the line as (P) and the one below as (Q).
  5. Use the ruler to join (P) and (Q). The line (PQ) is the perpendicular bisector of (AB).

What do we see?

  • The line (PQ) cuts (AB) exactly in the middle. The mid-point is the centre (O).
  • The line (PQ) also passes through (O) and continues to meet the circle at two opposite points. Mark these points as (C) and (D).

Why is (PQ) another diameter?

In a circle, any straight line that passes through the centre and meets the circle at two points is a diameter.

(PQ) is the perpendicular bisector of (AB).)

(Rightarrow PQ ext{passes through the midpoint of } AB.)

( ext{Midpoint of } AB ext{ is the centre } O.)

(Rightarrow PQ ext{ passes through } O.)

(Rightarrow PQ ext{ meets the circle at two opposite points } C,D.)

( herefore CD ext{ (the same line as } PQ) ext{ is a diameter.})

Final Answer

Yes. The perpendicular bisector of a diameter passes through the centre of the circle and therefore forms another diameter of the circle.

Tips

  • Keep the compass width fixed while drawing intersecting arcs.
  • Label points clearly so you can follow each step without confusion.
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