Maharashtra Board Class 10 Maths Part II Chapter 4 Geometric Constructions PDF Download

Geometric Constructions

Let's Study

Construction of a triangle similar to the given triangle

To construct a triangle, similar to the given triangle, bearing the given ratio with the sides of the given triangle.

(i) When vertices are distinct

(ii) When one vertex is common

Construction of a tangent to a circle.

To construct a tangent at a point on the circle.

(i) Using centre of the circle.

(ii) Without using the centre of the circle.

To construct tangents to the given circle from a point outside the circle.

Let's Recall

In the previous standard you have learnt the following constructions. Let us recall those constructions.

To construct a line parallel to a given line and passing through a given point outside the line.

To construct the perpendicular bisector of a given line segment.

To construct a triangle whose sides are given.

To divide a given line segment into given number of equal parts

To divide a line segment in the given ratio.

To construct an angle congruent to the given angle.

In the ninth standard you have carried out the activity of preparing a map of surroundings of your school. Before constructing a building we make its plan. The surroundings of a school and its map, the building and its plan are similar to each other. We need to draw similar figures in Geography, architecture, machine drawing etc. A triangle is the simplest closed figure. We shall learn how to construct a triangle similar to the given triangle.

Teacher's Note

When you make a map of your school, you draw it smaller than real size. This is similar figure. Like this, maps, building plans, and machine drawings all use similar shapes to show real things in smaller size.

Exam Trick

Remember: Similar figures have the same shape but different sizes. The angles are all equal, but the sides are in proportion. If you know the ratio, you can find all side lengths.

Points to Remember

Similar triangles have equal angles and sides in the same ratio.
You can make smaller or bigger similar figures using proportions.
Maps and building plans are examples of similar figures.
A triangle is the simplest closed figure to construct.

Construction of Similar Triangle

To construct a triangle similar to the given triangle, satisfying the condition of given ratio of corresponding sides.

The corresponding sides of similar triangles are in the same proportion and the corresponding angles of these triangles are equal. Using this property, a triangle which is similar to the given triangle can be constructed.

Ex. (1) Triangle ABC ~ Triangle PQR, in Triangle ABC, AB = 5.4 cm, BC = 4.2 cm, AC = 6.0 cm. AB: PQ = 3:2. Construct Triangle ABC and Triangle PQR.

Construct Triangle ABC of given measure.

Triangle ABC and Triangle PQR are similar.

Therefore their corresponding sides are proportional.

\[\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = \frac{3}{2}\]

As the sides AB, BC, AC are known, we can find the lengths of sides PQ, QR, PR.

Using equation above:

\[\frac{5.4}{PQ} = \frac{4.2}{QR} = \frac{6.0}{PR} = \frac{3}{2}\]

Therefore PQ = 3.6 cm, QR = 2.8 cm and PR = 4.0 cm

If we know the lengths of all sides of Triangle PQR, we can construct Triangle PQR.

In the above example (1) there was no common vertex in the given triangle and the triangle to be constructed. If there is a common vertex, it is convenient to follow the method in the following example.

Ex.(2) Construct any Triangle ABC. Construct Triangle A'BC' such that AB : A'B = 5:3 and Triangle ABC ~ Triangle A'BC'

Analysis : As shown in the figure, let the points B, A, A' and B, C, C' be collinear.

Triangle ABC ~ Triangle A'BC' therefore Angle ABC = Angle A'BC'

\[\frac{AB}{A'B} = \frac{BC}{BC'} = \frac{AC}{A'C'} = \frac{5}{3}\]

Therefore sides of Triangle ABC are longer than corresponding sides of Triangle A'BC'.

Therefore the length of side BC' will be equal to 3 parts out of 5 equal parts of side BC. So if we construct Triangle ABC, point C' will be on the side BC, at a distance equal to 3 parts from B. Now A' is the point of intersection of AB and a line through C', parallel to CA.

Teacher's Note

When you draw a similar triangle that is smaller, you place point C' on the side BC at distance equal to 3 parts from point B. This works like making a smaller copy of a picture.

Exam Trick

Remember: In similar triangles, divide one side into equal parts. Mark the required point and draw a parallel line. This parallel line will give you the other vertex. Use Basic Proportionality Theorem to check your answer.

Points to Remember

Corresponding sides of similar triangles are in the same ratio.
Draw a line parallel to one side to create a similar triangle.
Divide the side into equal parts based on the ratio given.
The parallel line cuts the other sides proportionally.

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