RS Aggarwal Solutions for Class 6 Chapter 14 Constructions

Exercise 14.1

Question 1. Explain the following: (i) Circle (ii) Radius (iii) Centre (iv) Diameter (v) Chord (vi) Interior of a circle
Answer:
(i) A circle consists of all points on a plane where the distance from a fixed point stays the same.
(ii) The radius of a circle is a line segment that goes from its centre to a point on the circle. It represents the constant distance between all points on the circle and its centre.
(iii) The centre of a circle is the fixed point from which all points remain at an equal distance.
(iv) A diameter of a circle is a line segment that goes across the centre, with both end points on the circle.
(v) A chord of a circle is a line segment whose two end points lie on the circle.
(vi) The interior of a circle is the set of all points that sit inside the circle.
In simple words: A circle is a round shape where every point is equally far from the middle. The radius is the distance from the middle to the edge. The diameter is a line across the whole circle through the middle. A chord is any line segment connecting two points on the circle.

Exam Tip: Memorize that radius = half of diameter and that a diameter is always a chord, but not all chords are diameters.

Question 2. Take a point on your notebook and draw circle of radii 4 cm, 3 cm and 6.5 cm, each having the same centre 0.
Answer: The diagram below displays three circles with radii of 4 cm, 3 cm, and 6.5 cm respectively, all sharing the same centre point O. These are concentric circles, where each circle is drawn from the same central location but has a different radius value.

Exam Tip: When drawing concentric circles, use a compass and keep the centre point fixed while changing only the radius measurement for each circle.

Question 3. Draw a circle with centre o and any radius. Draw AC and BC two perpendicular diameters of the circle. Join AB, BC, CD and DA.
Answer: The diagram shown below displays a circle with centre O, where AC and BC are two perpendicular diameters. When we connect the four endpoints A, B, C, and D, we form a square inscribed in the circle. This square has all sides equal and all angles at 90 degrees.

Exam Tip: When two perpendicular diameters are drawn and their endpoints are joined, they always create a square inside the circle.

Question 4. Draw a circle with centre 0 and radius 6 cm. Mark points P, Q, R such that (i) P lies on the circle, (ii) Q lies in the interior of the circle, and (iii) R lies in the exterior of the circle. Rewrite each of the following statements using the correct symbol (=, < or >):
(i) OO....5 cm
(ii) OR...5 cm
(iii) OR ....5 cm
Answer:
The diagram below shows the points P, Q and R positioned as described:
(i) P lies on the circle: OP = 6 cm
(ii) Q lies in the interior of the circle: OQ < 6 cm
(iii) R lies on the exterior of the circle: OR > 6 cm
In simple words: When a point sits on the circle's edge, its distance from the centre equals the radius. If it's inside, the distance is smaller. If it's outside, the distance is bigger.

Exam Tip: Use the radius as your reference point: equal distance means on the circle, less means inside, more means outside.

Question 5. Take two points A and B on the page of your note book. Draw a circle with centre A which passes through B.
Answer: The diagram shown below displays a circle with centre at point A. Since the circle passes through point B, the distance AB becomes the radius of the circle. To create this, place the compass point at A, adjust it so the pencil reaches B, and then draw the full circle.

Exam Tip: The radius of the circle is simply the straight-line distance from the centre A to the point B through which it passes.

Question 6. Draw a semi-circle with centre O and radius 5 cm. Is the diameter that determines the semi-circle, a part of the semi-circle?
Answer: The semi-circle with centre O and radius 5 cm is shown below. A diameter of a circle splits it into two equal parts, each called a semi-circle. The diameter itself is not part of the semi-circle but rather the boundary line that creates it. The semi-circle consists of the curved arc portion only, not the straight diameter line.
In simple words: The diameter is the straight line that divides the circle in half, but the semi-circle itself is just the curved part, so the diameter is not part of it.

Exam Tip: Remember that a semi-circle has a curved boundary (half the circumference) and a straight boundary (the diameter), but when we refer to the semi-circle as a region, we typically mean the area enclosed by both boundaries.

Question 7. The diameter of a circle is 14 cm, find its radius.
Answer: The radius of a circle is half the length of its diameter. Therefore, radius = diameter/2. Since diameter = 14 cm, radius = 14/2 = 7 cm.
In simple words: The radius is always half of the diameter. So if the diameter is 14 cm, just divide by 2 to get 7 cm.

Exam Tip: Always remember the relationship: diameter = 2 × radius, or radius = diameter ÷ 2. This is the most tested concept about circles.

Question 8. Given a circle with centre O and radius 2.5 cm, what is the length of the longest chord of the circle.
Answer: The longest chord in any circle is its diameter. Since the diameter equals twice the radius, the longest chord = 2 × 2.5 = 5 cm.
In simple words: The biggest chord you can draw in a circle is the diameter, which goes straight across through the middle. It's always twice as long as the radius.

Exam Tip: No matter what circle you're working with, the diameter will always be the longest possible chord. Any other chord will be shorter.

Question 9. Fill in the blanks:
(i) The diameter of a circle is ..... times its radius.
(ii) The diameter of a circle is the ..... chord of the circle.
(iii) The diameter of a circle pass through ........
(iv) A chord of a circle is a line segment with its end points on the.......
(v) If join any two points on a circle by a line segment, we obtain...... of the circle.
(vi) A radius of a circle is a line segment with one end at ...... and at......
(vii) All radii of a circle are......
(viii) The diameters of a circle are ......
(ix) The total number of diameters of a circle is .......
(x) Every point on a circle is ....... from its centre.
(xi) A chord of a circle contains exactly ....... points of the circle.
(xii) A diameter is the longest .......
(xiii) Concentric circles are circles having ......
Answer:
(i) two
(ii) longest
(iii) the centre of the circle
(iv) circle
(v) chord
(vi) the centre, on the circle
(vii) equal
(viii) concurrent
(ix) infinite
(x) equidistant
(xi) two
(xii) chord
(xiii) the same centre point

Exam Tip: These fill-in-the-blanks cover all the key definitions and properties of circles. Learn each one carefully as they form the foundation of circle geometry.

Question 10. In each of the following, state if the statement is true (T) of false (F):
(i) Every circle has a centre.
(ii) The centre of a circle is a point of the circle.
(iii) Any two radii of a circle make up a diameter.
(iv) Every chord of a circle is parallel to some diameter of the circle.
(v) A circle is symmetric about each of its diameters.
(vi) The diameter is twice the radius.
(vii) A radius is a chord of the circle.
(viii) Concentric circles have the same radii.
(ix) The nearer a chord to the centre of a circle, the longer is its length.
Answer:
(i) T
(ii) F
(iii) F
(iv) F
(v) T
(vi) T
(vii) F
(viii) F
(ix) T
In simple words: A circle always has a fixed middle point, but the centre itself is not on the circle's edge. The diameter is always double the radius. A chord gets longer when it's closer to the centre. Only some of these statements are correct.

Exam Tip: Pay special attention to statements (ii), (iii), (iv), (vii), and (viii) - these are common traps that test whether you understand the precise definitions of circle terms.