CBSE Class 10 Mathematics Constructions Notes Set 01

Class X

Chapter: Constructions

Top Definitions

1. The ratio of the side of the triangle to be constructed with the corresponding sides of the given triangle is known as their scale factor.

2. Reduced scale factor figures are the geometric figure to be constructed is smaller in size.

3. Enlarged scale factor figures are the geometric figure to be constructed is larger in size.

 Top Concepts

1. To divide a line segment internally in a given ratio m: n, where both m and n are positive integers, we follow the following steps:

Step 1: Draw a line segment AB of given length by using a ruler.

Step 2: Draw any ray AX making an acute angle with AB.

Step 3: Along AX mark off (m + n) points A₁, A₂,………A_M, A_m+1,………,A_m+n, such that AA₁ = A₁A₂ = Am_+n-1 A_m+n.

Step 4: Join B A_m+n

Step 5: Through the point Am draw a line parallel to A_m+n B by making an angle equal to ÐAA_m+n B at A_m.

Suppose this line meets AB at point P.

The point P so obtained is the required point which divides AB internally in the ratio m: n.

2. Constructions of triangles similar to a given triangle:

(a) Steps of constructions when m < n:

Step 1: Construct the given triangle ABC by using the given data.

Step 2: Take any one of the three side of the given triangle as base. Let AB be the base of the given triangle.

Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX below the base AB.

Step 4: Along AX mark off n points A1, A2, A3,………, An such that

AA₁ = A₁A₂ = ……… = A_n-1 A_n

Step 5: Join A_nB

Step 6: Draw A_mB’ parallel to A_nB which meets AB at B’.

Step 7: From B’ draw B’C’||CB meeting AC at C’.

Triangle AB’C’ is the required triangle each of whose sides is ( m / n )^th of the corresponding side of ΔABC.

(b) Steps of construction when m > n:

Step 1: Construct the given triangle by using the given data.

Step 2: Take any one of the three sides of the given triangle and consider it as the base. Let AB be the base of the given triangle.

Step 3: At one end, say A, of base AB. Construct an acute angle ∠BAX below base AB i.e., con the opposite side of the vertex C.

Step 4: Along AX mark off m (large of m and n) points A₁, A₂, A₃,………A_n of AX such that AA₁ = A₁A₂ = ………= A_m-1A_m.

Step 5: Join A_nB to B and draw a line through A_m parallel to A_nB, intersecting the extended line segment AB at B’.

Step 6: Draw a line through B’ parallel to BC intersecting by the extended line segment AC at C’.

Step 7: ΔAB’C’ so obtained is the required triangle.

3. Constructions of tangent to a circle:

a. To draw the tangent to a circle at a given point on it, when the centre of the circle is known.

Given : A circle with centre O and a point P on it.

Required : To draw the tangent to the circle at P.

Steps of construction:

i. Join OP

ii. Draw a line AB perpendicular to OP at the point P. APB is the required tangent at P.

4. To draw the tangent to a circle at a given point on it, when the centre of the circle is not known.

Given : A circle and point on it.

Required : To draw the tangent to the circle at P.

Steps of construction:

i. Draw any chord PQ and join P and Q to a point R in major arc PQ (or minor arc PQ).

ii. Draw ∠QPB equal to ∠PRQ and on opposite side of chord PQ. The line BPA will be a tangent to the circle at P.

5. To draw the tangent to a circle from a point outside it (external point) when its center is known.

Given : A circle with center O and a point P outside it.

Required : to construct the tangents to the circle from P.

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