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Class 10 Mathematics Revision Notes for Constructions of Chapter 11:
Construction is one of the most important chapters in the CBSE class 10 mathematics subject. Unlike other chapters where students can understand and learn the formulas, this chapter makes use of certain construction methods and steps to solve questions. Hence, it is essential for students to thoroughly revise every step and method to perform well in the examination. Extramarks has provided Class 10 Chapter 11 Mathematics Notes on its official website. The notes have been prepared to adhere to the CBSE Syllabus and NCERT Books. In addition to these notes, students can also access other study material such as CBSE Sample Papers, Important Questions, and CBSE Previous Years question papers.
Class 10 Mathematics Revision Notes for Constructions of Chapter 11 – Free PDF Download
Access Class 10 Mathematics Chapter 11 – Constructions:
Division of a Line Segment:
1. Use and purpose:
 To divide a line in a given ratio.
 When there is no precise way of measuring and dividing the line segment with scale.
2. Construction procedure:
Bisecting a Line Segment:
 Step 1: With a radius of more than half the length of the line segment, draw arcs centered at either end of the line segment (from points A and B) so that they intersect on either side of the line segment (like P and Q in the figure given below).
 Step 2: The line segment is bisected by the line segment on joining the points of intersection.
 Construction of a line segment in a given ratio:
Solved example with the solution for practice:
Given a line segment AB, divide it in the ratio m:n, where both m and n are positive integers. Suppose we want to divide AB in the ratio of 3:2 (m=3, n=2).
Solution:
Step 1: Draw any ray AX, making an acute angle with line segment AB.
Step 2: Locate 5 (= m + n) points A1, A2, A3, A4 and A5 on AX such that AA1 = A1A2 = A2A3 = A3A4 = A4A5.
Step 3: Join BA5. (A(m+n)=A5)
Step 4: Through the point A3 (m=3), draw a line parallel to BA5 (by making an angle equal to ∠AA5B ) at A3 intersecting AB at point C.
 Diagram:
To Construct a Triangle Similar To a Given Triangle as Per the Given Scale Factor:
1. Two cases:
 The triangle to be constructed is smaller than the given triangle.
 The triangle to be constructed is larger than the given triangle.
Note: The same methods would apply to the general case also.
2. Construction procedure:
Solved example with the solution for practice:
Suppose we want to construct a triangle whose sides are 3/4 times the corresponding sides of a given triangle. BB3/BB4 = ¾
Solution:
Steps of Construction:
Step 1: Draw any ray BX making an acute angle with side BC (on the side opposite to the vertex A).
Step 2: Mark 4 consecutive distances(since the denominator of the required ratio is 4) on BX as shown.
Step 3: Join B4C as shown in the figure.
Step 4: Draw a line through B3 parallel to B4C to intersect BC at C’.
Step 5:Draw a line through C’ parallel to AC to intersect AB at A’. ΔA′BC′ is the required triangle.
 Diagram:
Construction of Tangents to a Circle:
1. Definition of a Tangent: A tangent to a circle is a line that touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.
 If the point is in an interior region of the circle, any line through that point will be a secant. So, in this case, there is no tangent to the circle.
 When the point lies on the circle, there is accurately only one tangent to a circle.
 When the point lies outside of the circle, there are exactly two tangents to a circle.
2. Construction procedure:
To construct the tangents to a circle from a point outside it.
 Consider a circle with center O and let P be the exterior point from which the tangents are to be drawn.
Steps of Construction:
Step 1: Join the PO and bisect it. Let M be the midpoint of PO.
Step 2: Taking M as the center and MO (or MP) as the radius, draw a circle. Let it intersect the given circle at the points Q and R.
Step 3: Join PQ and PR.
Step 4: PQ and PR are the required tangents to the circle.
 Diagram:
Summary:
Construction is one such chapter in Class 10 Mathematics which requires a lot of inhand practice and solving the questions. This chapter provides a good opportunity for students to gain marks easily in the exam as the questions are of a similar nature to the examples. Class 10 Mathematics Chapter 11 Notes by Extramarks have been simplified to the finest level for students to prepare well and have a grip over all the concepts related to the Constructions chapter.
FAQs (Frequently Asked Questions)
1. What is meant by the scale factor in constructions?
The scale factor means the ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle.
2. How to construct the tangents of a circle if the center of the circle is not given?
If the center of the circle is not given, you may locate its center first by taking any two nonparallel chords and then finding the point of intersection of their perpendicular bisectors, and then following the steps for the construction of the tangents.
3. What is a secant? How is it different from tangent?
A tangent to a circle is a line that touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it whereas a straight line that intersects a circle in two points is called a secant line.