CBSE Grade 10 Maths Chapter 11

Construction is very important chapter of geometry. One of the aims of the studying construction is to acquire the skill of drawing figures accurately.

In this chapter we will learn how to divide a line into line segments, construction of triangles using scale factor, construction of tangents to a circle with two different cases.

Dividing a Line Segment

Steps for Bisecting a Line Segment

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

Step 2: Join the points of intersection. The line segment AB is bisected by the line segment PQ joining the points of intersection. Hence PQ is the perpendicular bisector of AB .

Example : 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 $A_{1}, A_{2}, A_{3}, A_{4} a n d A_{5}$ 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 the point $A_{3} (m = 3)$ , draw a line parallel to $B A_{5}$ (by making an angle equal to $∠ A A_{5} B)$ at $A_{3}$ intersecting AB at the point C.

Then, AC : CB = 3 : 2.

Constructing Similar Triangles

Steps for 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

Step 1: Draw any ray $B X$ making an acute angle with side $B C$ (on the side opposite to the vertex A).

Step 2: Mark 4 consecutive distances(since the denominator of the required ratio is 4) on $B X$ as shown.

Step 3: Join $B_{4} C$ 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’. $Δ A^{'} B C^{'}$ is the required triangle.

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

Tangents

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.
In the diagram given below, PQ is the tangent, touching the circle at A.

Number of Tangents to a circle from a given point

Case A: Point is in the interior region of the circle

If the point 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.
In the diagram given below, AB is a secant drawn through the point S

Case B: Point is on the circle

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

Case C: Point is in the exterior region of the circle

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

PT1 and PT2 are tangents touching the circle at T1 and T2

Drawing Tangents to a Circle

Case A: Drawing tangents to a circle from a point outside the circle

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 radius, draw a circle. Let it intersect the given circle at the points Q and R.

Step 3: Join PQ and PR

PQ and PR are the required tangents to the circle.

Case B: 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.

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