CBSE Class 6 Mathematics Basic Geometrical Ideas Assignment Set 03

Question 1. Explain the following terms by drawing a diagram
(a) Circle (b) radius (c) diameter (d) chord
(e) Biggest chord (f) secant (g) interior of the circle (h) Exterior of the circle
(i) arc of a circle (j) Segment
(k) Semi-circle (l) sector (m) centre
Answer:
(a) Circle: A circle is a round shape. Every point on its boundary is at the same distance from the middle point.
(b) Radius: This is a straight line from the center to any point on the boundary of the circle.
(c) Diameter: This is a straight line that goes from one side of the circle to the other, passing through the center.
(d) Chord: A straight line that joins any two points on the circle's boundary.
(e) Biggest chord: The diameter is the largest possible chord in any circle.
(f) Secant: A line that goes through the circle and cuts it at two different points.
(g) Interior of the circle: This is the space inside the boundary of the circle.
(h) Exterior of the circle: This is the space outside the boundary of the circle.
(i) Arc of a circle: A portion of the outer boundary of the circle.
(j) Segment: The area inside a circle between a chord and an arc.
(k) Semi-circle: Half of a full circle. It is made when a diameter cuts the circle in two equal parts.
(l) Sector: The area inside a circle between two radii and an arc. It looks like a slice of pie.
(m) Centre: The exact middle point of a circle. All points on the circle are at an equal distance from this point
In simple words: A circle is a round loop. Inside, it has parts like the center, radius (line to the edge), and diameter (line all the way across). We can also have pizza-slice shapes called sectors and cut-off shapes called segments.

Exam Tip: Remember that the diameter is always the longest chord in any circle, and its length is exactly twice the radius.

Question 2. With the help of compass draw too circle of radius 3.5cm and of 5cm.
Answer:
Here are the simple steps to draw these circles:
1. Use a scale to open your compass so the gap between the needle and pencil point is exactly 3.5 cm.
2. Place the needle on the paper and turn the compass to draw the first circle.
3. Now, open the compass wider so the gap is exactly 5 cm on the ruler.
4. Place the needle at a different spot on the paper and draw the second circle.

In simple words: We use a ruler to set the size on a compass. Then we rotate it on paper to draw two circles of different sizes.

Exam Tip: Keep the metal tip of the compass firmly pressed on the paper so that the circle does not lose its shape while drawing.

Question 3. Draw any circle of any radius. Draw a line passing through the centre and name it as AB Take any point C on the circumference. Measure \( \angle ACB \)
Answer: The angle \( \angle ACB \) will always measure \( 90^\circ \). The line AB is the diameter of the circle because it goes through the center. Any angle made on the boundary of a circle by connecting the two ends of the diameter is always a right angle.

In simple words: When you draw a triangle inside a circle using the middle line (diameter) as the base, the top corner will always form a perfect square corner (90 degrees).

Exam Tip: Remember the theorem: "The angle subtended by a diameter in a semicircle is always a right angle (\( 90^\circ \))."

Question 4. Draw any circle of any radius. Take any three points A, B and C such that:
(a) point A lies on the circle
(b) point B in the interior of the circle
(c) point C in the exterior of the circle.
Answer:
(a) Point A is placed directly on the curved boundary of the circle.
(b) Point B is placed anywhere inside the circle.
(c) Point C is placed anywhere outside the circle.

In simple words: Put point A on the line of the circle, point B inside the circle, and point C outside the circle.

Exam Tip: Ensure that point A is exactly on the boundary, not slightly inside or outside, to get full marks for position questions.

Question 5. Draw a circle of r = 4cm. Draw a chord AB of the circle. Take two points P and Q in such a way that
(a) arc APB is minor arc
(b) arc AQB is major arc
Answer:
A chord AB divides the circle's boundary into two curves or arcs:
1. The smaller curve is called the minor arc. We place point P on this smaller curve to form the minor arc APB.
2. The larger curve is called the major arc. We place point Q on this larger curve to form the major arc AQB.

In simple words: A line cut (chord) splits a circle into a small piece and a big piece. Put P on the smaller outer curve and Q on the larger outer curve.

Exam Tip: A chord always divides a circle into two arcs. Unless the chord is a diameter, one arc is always smaller (minor) and the other is larger (major).

Question 6. Draw a circle of radius 3.5cm. make a sector in such a way that its angle is 60°.
Answer:
Follow these steps to draw the sector:
1. Draw a circle with a radius of 3.5 cm using a compass.
2. Draw a straight line from the center to the edge. This is the first radius.
3. Place a protractor on this radius line, with its center at the circle's center.
4. Mark a point at \( 60^\circ \) and draw a second radius line to that point.
5. The region enclosed between these two lines and the arc is the required sector.

In simple words: Draw a circle, then draw two lines from the center that are open like a \( 60^\circ \) slice of pizza. Shade that slice.

Exam Tip: Always use a sharp pencil and a protractor to measure the angle exactly. Label the center, radii, and the angle of the sector clearly.

Question 7. Draw two concentric circles of radius 4cm and 6cm.
Answer:
Concentric circles are circles that have the same center point but different sizes. Follow these steps to draw them:
1. Mark a point on the paper to be the common center.
2. Set your compass to 4 cm and draw the first circle around this center.
3. Without moving the center point, open your compass to 6 cm.
4. Draw the second circle around the same center.

In simple words: Concentric circles are circles that share the same middle point, like a target board. One is inside the other.

Exam Tip: When drawing concentric circles, make sure the steel point of the compass does not slip from the center when drawing the second circle.

Question 8. If radius of a circle is 5cm. Find the length of longest chord.
Answer: The longest chord of a circle is its diameter. The diameter is twice the length of the radius.
Given, Radius = 5 cm
Length of longest chord (Diameter) = \( 2 \times \text{Radius} \)
\( \implies \text{Length} = 2 \times 5\text{ cm} = 10\text{ cm} \)
Therefore, the length of the longest chord is 10 cm.
In simple words: The longest line you can draw across a circle is the diameter. It is always two times the radius. So, \( 2 \times 5\text{ cm} = 10\text{ cm} \).

Exam Tip: Always state the formula \( \text{Diameter} = 2 \times \text{Radius} \) before writing the final calculation to secure step-marking marks.

Question 9. Take a line AB of 5cm. Draw two circles at point A and B so that they touch together.
Answer: Since the two circles touch each other along the line segment AB of length 5 cm, the sum of their radii must equal 5 cm. If we make both circles the same size, each circle will have a radius equal to half of the line's length:
Radius of each circle = \( \frac{5\text{ cm}}{2} = 2.5\text{ cm} \)
So, we can draw two circles of radius 2.5 cm with centers at A and B respectively.

In simple words: The distance between the centers is 5 cm. If the two circles touch in the exact middle, each circle must have a radius of half of 5 cm, which is 2.5 cm.

Exam Tip: When two circles touch externally, the distance between their centers is always equal to the sum of their radii (\( d = r_1 + r_2 \)).

Question 10. Draw a circle of radius 4cm. note its circumference. Can you read the number of points on the circumference? If yes, write the number.
Answer: No, it is not possible to count the points on the circumference. This is because any curved line, including the boundary of a circle, is made up of an endless number of points. There are infinite points along the circumference.
In simple words: No, we cannot count them. The outline of a circle is made of a continuous line of tiny points that never ends. There are infinite points.

Exam Tip: Remember that any line segment or curve is composed of an infinite number of points, regardless of how small or large it is.

Question 11. What is the difference between Segment and Sector?
Answer:
Here is the main difference between a segment and a sector of a circle:
1. Sector: A sector is a part of a circle bounded by two radii and an arc. It looks like a slice of pie or pizza.
2. Segment: A segment is a part of a circle bounded by a chord and an arc. It looks like a cut-off slice of a circle, and does not touch the center.

In simple words: A sector is a slice that goes all the way to the center of the circle, like a pizza slice. A segment is a slice cut off by a straight line, like a cut-off crust, and doesn't reach the center.

Exam Tip: A key difference to remember is that a sector contains the center of the circle as one of its corners, whereas a segment does not include the center.

Question 12. How many circles can be drawn from one point?
Answer: An infinite number of circles can be drawn from a single point. If the point is used as the center, we can draw endless circles of different sizes around it. Similarly, we can also draw endless circles of different sizes that pass through this single point.
In simple words: We can draw an endless number of circles of different sizes around a single point.

Exam Tip: Whether the point is the center of the circles or lies on their circumferences, you can always draw an infinite number of circles using that single point.