Constructions and Tilings
1. What is bisection in geometry?
Answer:
Bisection means dividing a line segment or any geometrical object into two identical parts.
Explanation: When a line segment is divided into two equal halves, the process is called bisection. The point where the division occurs is called the midpoint. This idea is used in many geometric constructions.
2. Why does the line joining points A and B become the perpendicular bisector of XY?
Answer:
Because A and B are at equal distances from X and Y.
Explanation:
Given:
AX = AY
BX = BY
Any point that is equally distant from X and Y lies on the perpendicular bisector of XY. Since both A and B satisfy this condition, the line AB becomes the perpendicular bisector of XY. It passes through the midpoint of XY and makes a 90° angle with it.
3. State the property of points lying on the perpendicular bisector of a line segment.
Answer:
Any point that is at the same distance from the two endpoints of a line segment lies on its perpendicular bisector.
Explanation:
If a point P satisfies:
then P must lie on the perpendicular bisector of XY. This property is the basis for constructing perpendicular bisectors using a compass and ruler.
4. How can you construct the perpendicular bisector of a line segment XY?
Answer:
Steps:
Draw line segment XY.
With X as centre, draw arcs above and below XY.
With the same radius and Y as centre, draw arcs cutting the previous arcs.
Mark the intersection points as A and B.
Join A and B.
AB is the required perpendicular bisector.
Explanation: Points A and B are equally distant from X and Y. Therefore, the line joining them must be the perpendicular bisector of XY.
5. Why is the compass-and-ruler method of finding a midpoint more accurate than measuring with a scale?
Answer:
Because it is based on exact geometric properties instead of measurement.
Explanation:
Measurements made using a scale may contain small errors. The compass-and-ruler construction uses the property of equal distances and therefore gives a geometrically exact midpoint.
6. How can a 90° angle be constructed at a given point O on a line?
Answer:
Steps:
Mark two points X and Y on the line such that O is the midpoint of XY.
Construct the perpendicular bisector of XY.
The bisector passes through O and is perpendicular to the line.
Explanation: Since O is the midpoint of XY, the perpendicular bisector of XY passes through O and forms a right angle of 90°.
7. How was a rope used in the Śulba-Sūtras to construct a perpendicular bisector?
Answer:
A rope was attached to the endpoints of the line segment, and its midpoint was pulled above and below the segment to obtain two points.
Explanation: The midpoint of the stretched rope remains equally distant from both endpoints. Therefore, the two marked positions lie on the perpendicular bisector. Joining them gives the required perpendicular bisector.
8. Why are triangles OBC and OAC congruent during angle bisection?
Answer:
Because they satisfy the SSS congruence condition.
Explanation:
OA = OB
AC = BC
OC is common
Therefore,
\(\triangle OBC \cong \triangle OAC\)Hence,
\(\angle BOC = \angle AOC\)and OC bisects the angle.
9. Describe the steps for constructing an angle bisector.
Answer:
Steps:
Mark points A and B on the two arms of the angle such that OA = OB.
With equal radius, draw arcs from A and B.
Let the arcs intersect at C.
Join O and C.
OC is the angle bisector.
Explanation: The construction creates two congruent triangles. Therefore, the angle is divided into two equal parts.
10. How can a 45° angle be constructed using only a ruler and compass?
Answer:
First construct a 90° angle and then bisect it.
Explanation:
Since:
the angle bisector divides the right angle into two equal angles of 45° each.
11. What is the basic idea behind copying an angle?
Answer:
The idea is to create a congruent triangle using the SSS condition.
Explanation: The distances between points on the original angle are copied exactly onto a new location. Since the corresponding triangles are congruent, the copied angle is equal to the original angle.
12. Why does the angle-copying method produce an exact copy of the given angle?
Answer:
Because the constructed triangles are congruent by SSS.
Explanation: All three corresponding sides are equal. Therefore, the corresponding angles are also equal. Hence, the copied angle is exactly the same as the original angle.
13. How is the idea of copying an angle used to construct parallel lines?
Answer:
Equal corresponding angles are constructed.
Explanation: A transversal intersects the given line. By copying the angle formed at one point to another point on the transversal, equal corresponding angles are obtained. Therefore, the two lines become parallel.
14. What geometric conditions are needed for the support lines of a trefoil arch?
Answer:
AB = CD
∠BAD = ∠CDA
Explanation: Equal lengths and equal angles ensure symmetry. Symmetry is necessary for creating a balanced and attractive arch design.
15. Why can six congruent equilateral triangles form a regular hexagon?
Answer:
Because six angles of 60° fit exactly around a point.
Explanation:
Each equilateral triangle has:
Six such angles give:
\(6 \times 60^\circ = 360^\circ\)Therefore, they fit perfectly without gaps or overlaps, forming a regular hexagon.
16. How can a 60° angle be constructed?
Answer:
Steps:
Draw a ray AX.
Draw an arc centred at A.
Mark point B where the arc cuts AX.
Using the same radius, draw an arc from B cutting the first arc at C.
Join AC.
Then:
\(\angle CAX = 60^\circ\)Explanation: AB = BC = AC, so triangle ABC is equilateral. Every angle of an equilateral triangle measures 60°.
17. Why is each interior angle of the regular hexagon formed by six equilateral triangles equal to 120°?
Answer:
Because two 60° angles meet at each vertex.
Explanation:
\(60^\circ + 60^\circ = 120^\circ\)Therefore, every interior angle of the regular hexagon is 120°.
18. What is tiling?
Answer:
Covering a region using shapes without gaps or overlaps is called tiling.
Explanation: The shapes used in a tiling fit together perfectly. No empty space is left, and the shapes do not overlap each other.
19. Why can a 5 × 7 grid not be tiled completely using 2 × 1 tiles?
Answer:
Because it contains an odd number of unit squares.
Explanation:
Number of squares:
Each tile covers:
\(2\)squares.
Since 35 is odd, the squares cannot be grouped into pairs. Therefore, complete tiling is impossible.
20. Why is the black-and-white colouring method useful in tiling problems?
Answer:
It helps determine whether a region can be tiled.
Explanation: Each 2 × 1 tile always covers:
one black square
one white square
Therefore, a tileable region must contain equal numbers of black and white squares. If the numbers are unequal, tiling is impossible. This method makes difficult tiling problems easier to analyse and solve.
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