Practical Geometry

Practical Geometry is all about construction of different shapes and sizes. It is one of the most important branches of geometry. There are various two-dimensional and three-dimensional shapes that are introduced to us. In practical geometry, we will learn to draw such shapes with proper dimensions.

In practical geometry, we are going to learn:

  • How to construct parallel lines?
  • How to construct:
    • Triangle, when three sides are known (SSS Criterion)
    • Triangle, when two sides and included angle are known (SAS Criterion)
    • Triangle, when two angles and the included side are known (ASA Criterion)
    • Right triangle, when one leg-length and hypotenuse are given (RHS Criterion)

Let us discuss how to construct these shapes step by step.

Practical Geometry – Construction of Parallel Lines

Two lines are said to be parallel if they do not intersect or meet each other at a single point. The parallel lines extend indefinitely in both directions. It is denoted by the symbol ‘||’. The real-life examples of parallel lines are railway tracks, the edges of a ruler, etc.

In practical geometry, we can draw a line parallel to another line using ruler and compasses, only. To learn how to draw parallel lines step by step, click here.

Practical Geometry – Construction of Triangle

A triangle is a closed polygon that has three sides and three angles. The properties of triangles that define them as a triangle are:

  • Sum of all three angles is equal to 180 degrees
  • Exterior angle is equal to the sum of interior opposite angles
  • Sum of lengths of two sides is greater than the third side
  • In the right triangle, the square of the hypotenuse side is equal to the sum of squares of the adjacent side and perpendicular side.

To construct a triangle, there are a few conditions that must be met based on the above properties, such as:

  • All three sides of a triangle are known
  • Two sides and included angle must be known
  • Two angles and included side must be known
  • For a right triangle, the hypotenuse and a leg of the triangle

Constructing triangle (SSS Criterion)

We can easily construct a triangle, if the three sides of the triangle are known to us.

  • We can take one side as the base of the triangle (hence, the two vertices of the triangle are known to us).
  • In the next step, using a ruler and compass, measure another side and draw an arc, above the base.
  • Again, using a rule and compass, measure the third side and cut the arc.
  • Thus, we have got the third vertex of the triangle.

Learn more: Constructing Triangles with SSS Congruence

Constructing triangle (SAS Criterion)

In SAS condition, a triangle can be constructed, when its two sides and included angle between the given two sides are known to us.

  • In the given two sides, we can take one side as the base of the triangle, thus the two vertices of the triangle are known to us.
  • Now, taking one vertex as the center and with the help of a rule and protractor, measure the given angle and mark the point, to join the vertex and the point.
  • Now, taking the same vertex and with the help of a compass measure the other given side of the triangle and draw an arc to get the third vertex.
  • Join the third side of the triangle.

Know all the steps with respective figures: Constructing Triangles, SAS

Constructing Triangle (ASA Criterion)

The ASA criterion of triangles is similar to the SAS criterion. In ASA, any two angles of the triangle and the side included between them are given to us. Based on this condition we need to construct the triangle now.

  • Like the other constructions of triangles, we have to draw the base of the triangle first, based on the given side.
  • Now take the protractor and put it on both the ends of the base-line (vertices of the triangle) and measure the two given angles, respectively.
  • Thus, by joining the points to the respective vertices and extending them, we can get the intersection of the two lines. This point will be the third vertex of the triangle.
  • Hence, a triangle is constructed.

Constructing Triangle (RHS Criterion)

As we all know, a right triangle is a triangle that has any one of its angles equal to 90 degrees. Now, as per RHS criterion, a right triangle can be constructed, when its hypotenuse side and any one of the leg-side (perpendicular or adjacent sides) are known to us.

Learn in detail: Construction of Right Angle Triangle

Practice Problems

  1. Draw a line ‘m’. Draw a perpendicular to ‘m’ at any point on ‘m’. On this perpendicular, choose a point ‘a’, 5 cm away from ‘m’. Through a, draw a line ‘l’ parallel to ‘m’.
  2. Construct a triangle ABC, given that AB = 6 cm, BC = 7 cm and AC = 5 cm. [Use SSS Criterion]
  3. Construct a triangle PQR, given that PQ = 4 cm, QR = 6.5 cm and ∠PQR = 45°. [Use SAS Criterion]
  4. Construct triangle XYZ if it is given that XY = 7 cm, measure of ∠ZXY = 100° and measure of ∠XYZ = 30°. [Use ASA Criterion]
  5. Construct a triangle LMN, right-angled at N, given that LN = 3 cm and MN = 5 cm. [Use RHS Criterion]

Frequently Asked Questions on Practical Geometry

What is practical geometry?

Practical geometry is the branch of mathematics that deals with constructions of geometrical shapes.

How can we draw a parallel line?

To draw a line parallel to another line, we should know the distance between the two parallel lines. Use the concept of equal alternate angles or equal corresponding angles in a transversal diagram to draw a line parallel to another line.

How to draw an angle in practical geometry?

To construct an angle, first we need to draw the line that will be one of the arms of the angle. Take any point and put the protractor on the line such that the center of the protractor coincides with the point on the line. And also the edges of the protractor should coincide with the line. Thus, by marking the angle on the edge of the protractor, we can construct the angle.

Are right angles 90 degrees?

A right angle is also called 90-degree angle.

How to practically draw a triangle?

There are various scenarios based on which we can construct triangles. They are SSS criterion, SAS criterion, ASA criterion and RHS criterion.

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