**Constructions**

__Constructions__

We will study

- How to divide a line segment in a given ratio
- Construct a triangle similar to given triangle
- Construct tangent to a circle

__Construction 1__**:**To divide a line segment in a given ratio

Let’s divide a line AB in ration 2:3

Steps:

- Draw any ray AX, making an acute angle with AB.
- Locate 5 points A1 to A5 on AX at equal intervals.
- Join BA5.
- Through the point A2 , draw a line parallel to BA5 intersecting AB at point C
- Then, AC : CB = 2 : 3.

__Alternate method to divide a line__

Let’s divide a line AB in ration 3:2

Steps:

- Draw Line AB
- Draw AX making an acute angle with AB.
- Draw ray BY parallel to AX
- Locate points A1, A2, A3 on AX & B1, B2 on BY such that AA1 = A1A2 = A2A3 = BB1 = B1B2.
- Join A3B2 to meet AB at a point C

__Construction 2__**:**To construct a triangle similar to a given triangle as per given scale factor.

Given a Δ, construct another triangle whose sides are ¾ of corresponding sides of triangle Δ.

Steps:

- Draw Δ
- Draw ray AX making an acute angle with AB
- Locate 4 equal points A1 to A4 on AX & Join A4B
- Draw line through A3 parallel to A4B to intersect AB at B′.
- Draw line through B′ parallel BC to intersect AC at C’

__Construction 3__**:**To construct the tangents to a circle from a point outside it.

Steps:

- Draw a circle with centre O & Take a point P outside circle.
- Join PO and bisect it. Let M be the midpoint of PO.
- Taking M as centre & MO as radius, draw circle. Let it intersect the given circle at Q and R.
- Join PQ and PR. PQ and PR are required 2 tangents