Constructions Solutions for RS Aggarwal Class 10 Chapter 13
FAQs on RS Aggarwal Class 10 Solutions - Constructions
1. Why are Vedantu's RS Aggarwal Solutions for Class 10 Constructions considered helpful for the 2025-26 exams?
Vedantu's RS Aggarwal Solutions for Class 10 Constructions are beneficial because they provide accurate and step-by-step solutions for every problem in the textbook. This helps students understand the precise method required for each construction, verify their own work, and build a strong foundational understanding of geometric principles, which is crucial for scoring well in exams.
2. What are the exact steps to construct tangents to a circle from an external point as per the RS Aggarwal method?
To construct tangents to a circle from an external point P, follow these steps:
Let the centre of the circle be O. Join the external point P to the centre O.
Find the midpoint of the line segment OP by constructing its perpendicular bisector. Let's call this midpoint M.
With M as the centre and MO (or MP) as the radius, draw a new circle. This circle will intersect the original circle at two points, say A and B.
Join PA and PB. These are the two required tangents to the circle from the external point P.
3. How do you correctly divide a line segment in a given ratio m:n using a compass and ruler?
To divide a line segment AB in the ratio m:n, the standard geometric construction involves these steps:
Draw a ray AX making an acute angle with the line segment AB.
Using a compass, mark off (m + n) equal arcs along the ray AX. Let these points be A1, A2, ..., A(m+n).
Join the last point, A(m+n), to the endpoint B of the line segment.
From the point Am, draw a line parallel to A(m+n)B, which intersects the original line segment AB at a point C. This point C divides AB in the required ratio m:n.
4. What is the mathematical justification for the method of constructing tangents from an external point?
The justification lies in a fundamental circle theorem. When we construct a second circle with the line segment OP (from the external point to the centre) as its diameter, any angle inscribed in this semicircle is 90°. The points where the two circles intersect (A and B) form a triangle (e.g., ΔOAP). In this triangle, ∠OAP is an angle in the semicircle of the second circle, so ∠OAP = 90°. This means OA (the radius) is perpendicular to PA. Since a line from the centre that is perpendicular to a line at the circumference makes that line a tangent, PA is a valid tangent.
5. In the construction for dividing a line segment, why does drawing a parallel line correctly create the required ratio?
This method works because of the Basic Proportionality Theorem (BPT) or Thales's Theorem. In the triangle ΔABA(m+n), the line drawn from Am (let's call it AmC) is parallel to the side A(m+n)B. According to the BPT, if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides proportionally. Therefore, AC/CB = AAm/AmA(m+n). Since AAm represents 'm' parts and AmA(m+n) represents 'n' parts, the line segment AB is successfully divided in the ratio m:n.
6. Is it possible to construct a tangent to a circle from a point located inside it? Explain your reasoning.
No, it is not possible to construct a tangent from a point inside a circle. A tangent is defined as a line that touches the circle at exactly one point. Any line drawn from an interior point will inevitably pass through the circle and intersect it at two distinct points. Such a line is called a secant, not a tangent. Therefore, a point must be on or outside the circle to have a tangent associated with it.
7. What are the most common errors students make while solving construction problems from RS Aggarwal?
Common errors in construction problems include:
Inaccurate Measurements: Using a blunt pencil or a loose compass leads to imprecise arcs and lines.
Incorrect Bisection: Failing to find the exact midpoint of a line segment when constructing a perpendicular bisector.
Not Showing Construction Arcs: Erasing the faint arcs used for construction. These arcs are part of the solution and demonstrate the method used.
Confusing Tangent with Secant: Extending the tangent line incorrectly so it cuts through the circle.
Following the step-by-step solutions helps in avoiding these common mistakes.
