Class 6 Solutions Chapter-14 Constructions by RS Aggarwal from Vedantu
FAQs on RS Aggarwal Class 6 Solutions Chapter-14 Constructions
1. How do you construct the perpendicular bisector of a line segment using a ruler and compasses, as shown in RS Aggarwal Class 6 Maths Chapter 14?
To construct a perpendicular bisector for a given line segment (e.g., AB), follow these steps precisely:
With A as the centre, use a compass to draw arcs on both sides of the line segment AB. The radius must be more than half the length of AB.
Keeping the same radius, now use B as the centre and draw two more arcs that intersect the first set of arcs at points C and D.
Join the points C and D using a ruler. The line CD is the required perpendicular bisector of the line segment AB, intersecting it at a 90° angle.
2. What is the correct method to bisect an angle using only a compass and a ruler?
The standard method for bisecting a given angle (e.g., ∠PQR) involves these steps:
Place the compass point at the vertex Q and draw an arc of any convenient radius that cuts the arms QP and QR at points A and B, respectively.
With A as the centre, draw an arc in the interior of the angle. The radius should be more than half the distance between A and B.
Using the same radius, place the compass at point B and draw another arc that intersects the previous one at a point S.
Draw a ray from the vertex Q passing through point S. This ray, QS, is the angle bisector, dividing ∠PQR into two equal angles.
3. What are the steps to construct a 90° angle at the initial point of a given ray?
Constructing a 90° angle involves creating a perpendicular. Following the solutions in RS Aggarwal, you can use these steps for a ray OA:
With O as the centre, draw an arc that cuts the ray OA at point P.
With P as the centre and the same radius, draw an arc that intersects the first arc at Q. This creates a 60° angle (∠QOA = 60°).
With Q as the centre and the same radius, draw another arc that intersects the initial arc at R. This marks 120° (∠ROA = 120°).
Now, bisect the 60° angle between Q and R. With Q and R as centres, draw two arcs with the same radius to intersect at point S.
Join O and S. The ray OS forms a 90° angle with the original ray OA.
4. Why does the standard method for constructing a perpendicular bisector actually work?
The method works because of the geometric properties of congruent triangles. When you draw arcs of the same radius from the endpoints (A and B) of a line segment, the intersection points (C and D) are equidistant from both A and B. This means AC = BC and AD = BD. By joining the points, you create two triangles (ΔADC and ΔBDC) that are congruent by the SSS (Side-Side-Side) rule. This congruence guarantees that the line CD not only cuts AB into two equal halves (bisects it) but also intersects it at a perfect 90° angle.
5. How can a student construct a 75° angle using only a ruler and compass, based on the concepts from this chapter?
A 75° angle is a composite angle. You can construct it by combining two basic constructions: a 90° angle and a 60° angle.
First, construct a 90° angle (let's call it ∠XOA) and a 60° angle (∠YOA) on the same ray OA.
The angle between the arms OX and OY is 90° - 60° = 30° (∠XOY).
Now, bisect this 30° angle (∠XOY). The bisector will create a 15° angle.
Adding this 15° to the 60° angle (60° + 15°) gives you the required 75° angle. This is a common higher-order thinking question based on the chapter's principles.
6. What is the difference between drawing a perpendicular to a line and constructing a perpendicular bisector?
While both constructions involve creating a 90° angle, their purpose is different. A perpendicular to a line can be drawn from any point, either on the line or outside it, and its only requirement is to form a right angle with the line. A perpendicular bisector, however, is more specific. It must not only be perpendicular (at 90°) to a line segment but must also pass through its exact midpoint, dividing the segment into two equal lengths.
7. When constructing an angle bisector, why is it crucial that the two arcs drawn from the sides of the angle intersect using the same radius?
The size of the first arc drawn from the vertex can be of any convenient radius. However, the next two arcs must be drawn with the same radius from the points where the first arc cut the angle's arms. This is crucial because it ensures the intersection point is equidistant from both arms of the angle. If you were to use different radii, the intersection point would be closer to one arm than the other, and the resulting line from the vertex would not be a true bisector that divides the angle into two perfectly equal halves.
