# Constructions Class 10 Notes: Chapter 11

According to the CBSE Syllabus 2023-24, this chapter has been removed fromΒ NCERT Class 10 Maths textbook.

## CBSE Class 10 Maths Construction Notes:-

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.

PQ is the perpendicular bisector of AB

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 segmentΒ AB.

Step 2: Locate 5 (= m + n) points A1,A2,A3,A4andA5 on AX such that AA1=A1A2=A2A3=A3A4=A4A5

Step 3: Join BA5.(A(m+n)=A5)

Step 4: Through point, A3(m=3), draw a line parallel to BA5 (by making an angle equal to β AA5B) at A3 intersecting AB at point C.

Then, AC : CB = 3 : 2.

Division of a line segment

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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 BCΒ  (on the side opposite to the vertex A).

Step 2: Mark 4 consecutive distances(since the denominator of the required ratio is 4) on BXΒ  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 ABΒ atΒ  A’. ΞAβ²BCβ² 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:Β  The triangle to be constructed is smaller than the given triangle, and the scale factor is less than 1.Β

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.

PQ is the tangent, touching the circle at A

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### Number of tangents to a circle from a given point

i) If the point is in anΒ interior region ofΒ the circle, any line through thatΒ point will be a secant. So, in this case, there is no tangent to the circle.

AB is a secant drawn through the point S

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

PQ is the tangent touching the circle at A

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

PT1 and PT2Β are tangents touching the circle at T1 and T2

### 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 theΒ PO and bisect it. Let M be the midpoint ofΒ 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:Β Join PQ and PR

Step 3:PQ andΒ 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

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

To learn more about the construction of angles, perpendicular bisectors, similar triangles, tangents to circles, etc., download BYJU’S – The Learning App today!