**Revision Notes on Constructions**

**Division of a Line Segment**

If we have to divide a line segment in particular ratio, then we can do it by measuring the length on the ruler and mark it on the line. But if we don’t have anything to measure then we can do it by using steps of construction.

**Method 1**

AB is a line segment of 4 cm. Divide it in the ratio of 1:3 using a compass.

**Steps of Construction**

**Step 1**: Draw a line AC of any length by making an acute angle with the given line segment AB.

**Step 2**: Using any small length on the compass mark 4 points of equal size on AC, so that AX_{1} = AX_{2} = AX_{3} = AX_{4}.We are marking 4 points as we have to divide the line in the ratio of 1:3, so 1 + 3 = 4.

**Step 3**: Now join BX_{4}.

**Step 4**: Draw a line from point X_{1} to line AB parallel to BX_{4}, which intersects AB at point P.

Now AP: PB = 1:3.

**Method 2(Alternative Method)**

A line can also be divided by another method.

**Steps of Construction**

**Step 1:** Draw a line AC with the acute angle with the line segment AB.

**Step 2:** Draw another line DB parallel to AC so that ∠BAX = ∠ADB

**Step 3:** Mark the points X_{1}(m = 1) on AC and Y_{1}, Y_{2}, Y_{3 }n = 3) on DB so that AX_{1 }= BY_{1 }= Y_{1}Y_{2 }= Y_{2}Y_{3}.

**Step 4:** Join X_{1}Y_{3} so that it intersects line AB at P.

AP:PB = 1:3

**Construction of a Triangle similar to a given Triangle as per given Scale**

The scale factor is the ratio of the sides of the triangle given to the sides of the triangle to be made by the steps of construction.

**Example:**

Draw a triangle similar to ∆ABC with its sides equal to 2/3 of the corresponding sides of the given triangle ABC. (Scale factor = 2/3).

**Steps of Construction**

**Step 1:** Draw a line AX by making an acute angle with the line segment AB.

**Step 2:** Mark three points of equal size using a compass on the line AX. Points will be depending upon the scale factor as we have to mark the number of points which is greater in the scale factor. In the ratio of 2/3(3 > 2).

**Step 3:** Join BX_{3} and draw a line from X_{2} parallel to BX_{3} to intersect AB at P.

**Step 4:** Draw a line parallel to BC from point P to intersect AC at Q.

**Now ∆APQ ~ ∆ABC.**

**Remark:** Here we have made a similar triangle which is smaller than the given triangle because the scale factor was 2/3. But if we have scale factor like 5/3 then we will make a bigger triangle then the given triangle by taking 5 points on the line).

**Construction of Tangents to a Circle**

**Tangent** is a line which intersects the circle at one point only at the outer of the circle. It is always perpendicular to the radius of the circle.

**Example:**

Construct the pair of tangents to the circle of radius 3 cm from the point which is 7 cm away from its centre, and measure their lengths also.

**Steps of Construction**

**Step 1**: Draw a circle of radius 3 cm by taking O as the centre.

**Step 2:** Mark a point P outside the circle at a distance of 7 cm from the centre O. Join OP.

**Step 3:** Bisect the line segment OP, so that the perpendicular bisector of OP intersects it at the point M.

**Step 4:** Now draw another circle by taking M as centre and MO as radius, which intersects the given circle at two points’ i.e. T and Tꞌ.

**Step 5:** Now join PT and PTꞌ which are the required tangents and measure the length of the tangents.

**The length of the tangents is PT = PTꞌ = 6.3 cm.**