Draw a circle in which a chord of 6 cm length stands at a distance of 3 cm from the centre.
(Hint: Is it a circumcircle of a suitable triangle?)
Step-by-step solution
To find: Construct such a circle
Idea: Radius = √(32 + 32) = 3√2 cm. To avoid measuring √18, use the hint: the circumcentre of a right triangle is the midpoint of its hypotenuse. If AB = 6 cm and BD ⟂ AB with BD = 6 cm, the midpoint O of AD is 3 cm from AB (half of BD).
- Draw AB = 6 cm. At B construct a right angle and mark D on it with BD = 6 cm. Join AD.1 mark
- Bisect AD (perpendicular bisector) to get its midpoint O. In a right triangle the circumcentre is the midpoint of the hypotenuse, so OA = OB = OD.1 mark
- With centre O and radius OA (= ½AD = ½ × 6√2 = 3√2 ≈ 4.24 cm), draw the circle. It passes through A and B, so AB = 6 cm is a chord.½ mark
- Check the distance: the perpendicular OM from O to AB lands at the midpoint of AB, and OM = ½BD = 3 cm (O is the midpoint of AD and OM ∥ DB). So the 6 cm chord is 3 cm from the centre ✓.½ mark
Check: Half-chord 3 cm and distance 3 cm: radius = √(32 + 32) = √18 ≈ 4.24 cm ✓, the same as ½AD.
Answer to write in the exam
Steps of construction:
1. Draw AB = 6 cm.
2. At B, draw BX ⟂ AB; cut BD = 6 cm on BX. Join AD.
3. Draw the perpendicular bisector of AD; it meets AD at O.
4. With centre O and radius OA, draw the circle.
Radius = ½AD = ½√(62 + 62) = 3√2 ≈ 4.24 cm; OM ⟂ AB, OM = ½BD = 3 cm
∴ AB is a chord of 6 cm at a distance of 3 cm from the centre O.
Common mistakes that cost marks
- Drawing a circle of radius 3 cm: then the chord at 3 cm from the centre would have length 0.
- Taking the radius as 3 + 3 = 6 cm instead of √(32 + 32).
- Placing the centre 3 cm from A instead of 3 cm from the chord (along the perpendicular bisector of AB).
How this can come in the exam
A chord of length 10 cm is at a distance of 5 cm from the centre of a circle. The radius is
- 5 cm
- 10 cm
- 5√2 cm
- 15 cm
Show answer
(C) 5√2 cm
Radius = √(52 + 52) = 5√2 ≈ 7.07 cm.
Try one yourself
Find the radius of a circle in which a chord of 8 cm is at a distance of 1.5 cm from the centre, and say how you would draw it.
Show answer
Radius = √(42 + 1.52) = √18.25 ≈ 4.27 cm. Draw AB = 8 cm, its perpendicular bisector, mark O on it 1.5 cm from AB, and draw the circle with centre O through A.
More questions like this
- Show that rectangle is the only parallelogram that can be inscribed in a circle.
- Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.
- Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?
- In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: “The centre of the circle lies on the angle bisector of ∠BAC”.
- Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.