Constructions Class 10 Notes: Chapter 11

Class 10 maths notes for the topic ‘Constructions’ are provided here. These notes help with the revision of concepts for the CBSE Class 10 Board Exams. In this article, we will discuss how to construct the division of the line segment, the construction of tringles using scale factor, and the construction of tangents to a circle with two different cases in detail. Go through the below article to learn more.

Dividing a Line Segment

Bisecting a Line Segment

Step 1: With a radius of more than half the length of the line segment, draw arcs centred at either end of the line segment so that they intersect on either side of the line segment.

Step 2: Join the points of intersection. The line segment is bisected by the line segment joining the points of intersection.

2) 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 3:2 (m=3, n=2)

Step 1: Draw any ray ${AX}^{}$, making an acute angle with line segment ${AB}^{}$.

Step 2: Locate 5 (= m + n) points $A_1,A_2,A_3,A_{4}and{A}_5$ on AX such that $A{A}_1=A_1{A}_2=A_2{A}_3=A_3{A}_4=A_4{A}_5$

Step 3: Join $B{A}_5.(A_{(m+n)}=A_5)$

Step 4: Through point, $A_3(m=3)$, draw a line parallel to $B{A}_5$ (by making an angle equal to $\angle A{A}_{5}B)$ at $A_3$ intersecting AB at point C.

Then, AC : CB = 3 : 2.

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Constructing Similar Triangles

Constructing a Similar Triangle with a Scale Factor

Suppose we want to construct a triangle whose sides are 3/4 times the corresponding sides of a given triangle.

BB₃/BB₄ = 3/4

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 $B_4\mathrm{C,}$ as shown in the figure.

Step 4: Draw a line through $B_3$ parallel to $B_{4}C$ to intersect BC at C’.

Step 5: Draw a line through C’ parallel to AC to intersect AB at  A’. $\Delta A^{\prime }B{C}^{\prime }$ is the required triangle.

The same procedure can be followed when the scale factor > 1.

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There are two cases while constructing a triangle similar to a given triangle as per the given scale factor. They are:

Case 1:  The triangle to be constructed is smaller than the given triangle, and the scale factor is less than 1. 

Case 2: The triangle to be constructed is larger than the given triangle, and the scale factor is greater than 1.

Here, the scale factor means the ratio of the triangle sides to be constructed with the corresponding sides of the given triangle.

Drawing Tangents to a Circle

Tangents: Definition

A tangent to a circle is a line which touches the circle at exactly one point.

For every point on the circle, there is a unique tangent passing through it.

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Number of tangents to a circle from a given point

i) 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.

ii) When the point lies on the circle, there is accurately only one tangent to a circle.

iii) When the point lies outside of the circle, there are exactly two tangents to a circle.

Drawing tangents to a circle from a point outside the circle

 

To construct the tangents to a circle from a point outside it.

Consider a circle with centre O and let P be the exterior point from which the tangents to be drawn.

Step 1: Join the ${PO}^{}$ and bisect it. Let M be the midpoint of ${PO}^{}$.

Step 2: Taking M as the centre 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 3:${PQ}^{}$ and ${PR}^{}$ are the required tangents to the circle.

Drawing tangents to a circle from a point on the circle

To draw a tangent to a circle through a point on it.

Step 1: Draw the radius of the circle through the required point.
Step 2: Draw a line perpendicular to the radius through this point. This will be tangent to the circle.

Related Articles

• NCERT Solutions for Class 10 Maths Chapter 11- Constructions
• RD Sharma Solutions for Class 10 Maths Chapter 11 Constructions
• NCERT Exemplar Class 10 Maths Solutions for Chapter 10 Constructions
• Class 10 Maths Chapter 11 Constructions MCQs
• Important Questions Class 10 Maths Chapter 11-Contructions

Watch the below video to understand the Construction of Tangents to the Circle

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