Constructions: Triangles, Tangent, Segments and Methods

The word ‘construction’ in geometry means to draw geometric figures such as lines and circles using only a compass and ruler or scale. It is important to note that you are not allowed to measure angles with a protractor, or measure lengths with a ruler in constructions. 

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Explanation

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Line Segments

• Bisection of a Line Segment: 

Bisecting a line segment means that we are dividing that line exactly into two halves. Therefore, the two halves of the line segment are said to be in the ratio 1:1. 

Construction Steps-

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.

• Division of a Line Segment in the ratio m:n:

Construction Steps-

Given a line segment AB, divide it in the ratio m:n, where both m and n are positive integers.

Step 1: Draw any ray AX, making an acute angle with line segment AB.

Step 2: Locate (m + n) points A₁, A₂, A₃, A₄, ....A_(m+n) on A_X such that AA₁=A₁A₂=A₂A₃=……. =A_(m+n-1)A_(m+n)

Step 3: Join BA_(m+n).

Step 4: Through the point Am, draw a line parallel to BA_(m+n) (by making an angle equal to ∠AA_(m+n)B at Am intersecting AB at the point C.

Then, AC: CB = m: n.

Triangles

Construction of Triangle with a scale factor m:n:

• When m

Construction Steps-

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

Step 2: Make an acute angle ∠BAX, below the base AB.

Step 3: Along A_X, mark n points A₁, A₂ …, A_n, such that AA₁ = A₁A₂ = … = A_m-1 A_m = … A_n-1 A_n.

Step 4: Join AnB.

Step 5: From A_m, draw A_mB’ parallel to A_nB, meeting AB at B’.

Step 6: From B’, draw B’C’ parallel to BC, meeting AC at C’.

Step 7: Triangle AB’C’ is the required triangle, each of whose sides is mn (m < n) of the corresponding sides of ΔABC

• When m>n (Increasing the size of the Triangle)

Construction Steps-

Step 1: Construct a ΔABC by using the given data.

Step 2: Make an acute angle ∠BAX, below the base AB. Extend AB to AY and AC to AZ.

Step 3: Along A_X, mark m points A₁, A₂ …, A_n, A_m, such that AA₁ = A₁A₂ = A₂A₃ = … = A_n-1 A_n = … = A_m-1 A_m

Step 4: Join AnB.

Step 5: From A_m, draw A_mB’ parallel to A_nB, meeting AY produced at B’.

Step 6: From B’, draw B’C’ parallel to BC, meeting AZ produced at C’.

Step 7: Triangle AB’C’ is the required triangle, each of whose sides is (mn) (m > n) of the corresponding sides of ΔABC.

Circles

Construction of Tangent to the Circle from a Point Outside the Circle:

• When Centre is unknown:

If the centre of the circle is not known, then we first find the centre of the circle by drawing two non-parallel chords of the circle. The point of intersection of perpendicular bisectors of these chords gives the centre of the circle. Then we can proceed as below:

Construction Steps-

Step 1: Draw a circle with centre O of the given radius.

Step 2: Take a given point A on the circle.

Step 3: Join OA.

Step 4: Construct ∠OAB = 90°.

Step 5: Produce BA to B’ to get BAB’ as the required tangent.

• When Centre is known:

Construction Steps-

Step 1: Draw a circle with centre O.

Step 2: Join the centre O to the given external point P.

Step 3: Draw a right bisector of OP to intersect OP at Q.

Step 4: Taking Q as the centre and OQ = PQ as radius, draw a circle to intersect the given circle at T and T’.

Step 5: Join PT and PT’ to get the required tangents as PT and PT’.

Construction of Tangent to the Circle from a Point on the Circle:

Construction Steps-

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.