Question 1

Q1: Construct at least 4 different angles. Draw their bisectors.

Accepted Answer

To Construct: Four different angles and their bisectors. General Steps for Angle Bisection: Let's assume we have an angle $\angle PQR$ with vertex at Q. 1.  Place the compass point at the vertex Q and draw an arc that intersects the two arms of the angle, QP and QR. Let the intersection points be A...

Question 2

Q2: Construct the 8-petalled figure shown in Fig. 6.5.

Accepted Answer

To Construct: An 8-petalled flower figure. Steps of Construction: 1.  Draw a horizontal line l and a vertical line m intersecting at a point O. This creates four $90^{\circ}$ angles. 2.  Bisect each of the four $90^{\circ}$ angles. This results in four new lines passing through O. Now you have eight...

Question 3

Q3: In Step 2 of angle bisection, if arcs of equal radius are drawn on the other side, as shown in the figure, will the line OC still be an angle bisector...

Accepted Answer

Answer: Yes, the line joining the vertex to the intersection point of arcs drawn on the other side will also be the angle bisector. Justification: Let the given angle be $\angle XOY$ with vertex O. 1.  First, we mark points A on ray OX and B on ray OY such that $OA = OB$. This is done by drawing an...

Question 4

Q4: What are the other angles that can be constructed using angle bisection? Can you construct $65.5^{\circ}$ angle?

Accepted Answer

Angles that can be constructed: Using a ruler and compass, we can construct some basic angles like $90^{\circ}$ (perpendicular lines) and $60^{\circ}$ (equilateral triangle). By applying the angle bisection method repeatedly, and by adding or subtracting these angles, we can construct many other ang...

Question 5

Q5: Come up with a method to construct the angle bisector using a rope.

Accepted Answer

To Construct: The bisector of a given angle $\angle XOY$ using a rope. Steps of Construction: 1.  The angle $\angle XOY$ is marked on the ground with pegs at the vertex O and at points on the arms OX and OY. 2.  Take a piece of rope. Use it to measure a distance from the vertex O along the arm OX. M...

Question 6

Q6: Construct the following figure. How do we construct the petals so that they are of the maximum possible size within a given square?

Accepted Answer

To Construct: A four-petaled figure within a square, where the petals are of maximum size. Steps of Construction: 1.  Construct a square ABCD using a ruler and compass. 2.  Find the midpoints of the four sides of the square. Let M be the midpoint of AB, N be the midpoint of BC, P be the midpoint of...

Question 7

Q1: Use support lines in Fig. 6.11 to construct a pointed arch. Make different arches, by changing the radius of the arcs.

Accepted Answer

To Construct: A pointed arch. Steps for a Basic Pointed Arch (Equilateral Arch): 1.  Draw a horizontal line segment XY. This will be the base or opening of the arch. 2.  Place the compass point at X. 3.  Set the compass radius to be equal to the length of the base, XY. 4.  Draw a long arc above the...

Question 8

Q2: Make your own arch designs.

Accepted Answer

To Construct: A custom arch design, for example, a Trefoil Arch. A trefoil arch has three lobes (foils). Steps of Construction: 1.  Define the base: Draw a horizontal line segment AB which represents the width of the arch. 2.  Divide the base: Divide the segment AB into two equal halves at point M....

Question 9

Q1: Justify why AB in Fig. 6.4 is the perpendicular bisector.

Accepted Answer

To Justify: The line AB constructed using the rope method is the perpendicular bisector of the line segment XY. Justification: The construction involves a rope of a fixed length, say $L$, with its ends fastened at points X and Y. 1.  Finding point A: The midpoint of the rope is pulled taut to a poin...

Question 10

Q2: Can you think of different methods to construct a $90^{\circ}$ angle at a given point on a line using a rope?

Accepted Answer

Answer: Yes, here are two methods to construct a $90^{\circ}$ angle at a given point O on a line using a rope. Method 1: Using the Perpendicular Bisector Principle 1.  Let the given point be O on a line l drawn on the ground. 2.  Use a rope to measure a certain length from O along the line l to a po...

Question 11

Q1: Construct 4 pairs of parallel lines in different orientations.

Accepted Answer

To Construct: Four pairs of parallel lines. General Method (using corresponding angles): 1.  Draw a line and a point: Draw a line m and choose a point P that is not on m. 2.  Draw a transversal: Draw a straight line t that passes through point P and intersects line m at a point Q. 3.  Copy the angle...

Question 12

Q2: Construct the following figure.

Accepted Answer

To Construct: A parallelogram with lines drawn through its vertices parallel to its diagonals. Steps of Construction: 1.  Construct the parallelogram:     a. Draw a line segment AB.     b. From point A, draw another line segment AD at any angle to AB.     c. Through point B, construct a line paralle...

Question 13

Q1: When constructing the perpendicular bisector, is it necessary to have the same radius for the arcs above and below XY? Explore this through constructi...

Accepted Answer

Answer: No, it is not necessary to use the same radius for the arcs above and below the line segment XY. Justification: The perpendicular bisector of a line segment XY is the line containing all points that are equidistant from X and Y. To define a line, we only need two distinct points. 1.  Finding...

Question 14

Q2: Is it necessary to construct the pairs of arcs above and below XY? Instead, can we construct both the pairs of arcs on the same side of XY? Explore th...

Accepted Answer

Answer: No, it is not necessary to construct the arcs on opposite sides of XY. Both pairs of arcs can be constructed on the same side. Justification: To construct the perpendicular bisector, we need two distinct points that are equidistant from X and Y. 1.  Finding the first point (A): Choose a radi...

Question 15

Q3: While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them ? Explore this through construction, and...

Accepted Answer

Answer: Yes, it is absolutely necessary to use the same radius for the two arcs that intersect to find a single point on the perpendicular bisector. Justification: A point lies on the perpendicular bisector of a segment XY if and only if it is equidistant from both X and Y. Let's say we are trying t...

Question 16

Q4: Recreate this design using only a ruler and compass -

Accepted Answer

To Construct: The described geometric design, which is a four-petal flower shape inside a square. Steps of Construction: 1.  Use a ruler to draw a horizontal line segment. Let's call its endpoints P and Q. 2.  Construct the perpendicular bisector of the segment PQ. Let the bisector be line l, and le...

Question 17

Q1: Construct the following figures: (a) A 6-pointed star (b) A curved, six-petaled flower (c) A pattern of seven intersecting circles (d) An interwoven p...

Accepted Answer

(a) Construction of a 6-pointed star: 1. Draw a circle with center O and any radius. 2. Pick a point A on the circumference. With the same radius, step off six points around the circle: A, B, C, D, E, F. 3. Join the points to form a regular hexagon ABCDEF. 4. Join alternate vertices to form an equil...

Question 18

Q2: Optical Illusion: Do you notice anything interesting about the following figure? How does this happen? Recreate this in your notebook.

Accepted Answer

Observation: The figure, known as the Café Wall Illusion, appears distorted. Although it is constructed from a grid of perfectly straight horizontal and vertical lines, the horizontal lines appear to be tilted or sloped, and the squares can look like they are bulging. How it happens: This optical il...

Question 19

Q3: Construct this figure. [Hint: Find the angles in this figure.]

Accepted Answer

To Construct: A five-pointed star (pentagram) inside a regular pentagon. Note: The exact construction of a regular pentagon with only a ruler and compass is an advanced geometric procedure not typically covered at this level. The textbook itself mentions that this topic will be discussed in later ye...

Question 20

Q4: Draw a line $l$ and mark a point P anywhere outside the line. Construct a perpendicular to the given line $l$ through P.

Accepted Answer

To Construct: A perpendicular to a line l from an external point P. Steps of Construction: 1.  Given a line l and a point P not on l. 2.  Place the compass point at P. 3.  Open the compass to a radius that is large enough to cross the line l at two distinct points. Draw an arc that intersects line l...

NCERT Solutions