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The way of constructing a triangle with a compass depends on the information given in the question. The construction of triangles is very important while applying the Pythagorean theorem and trigonometry.

Here, we will learn how to construct a triangle if we have the following data.

All the three sides of a triangle are given.

The measure of the hypotenuse and one side is given in the right triangle.

Two sides of a triangle and included angle are given.

Two angles of a triangle and included sides are given.

Triangle is a two-dimensional polygon shape with three sides and three angles, which can be formed by joining the points in a plane.

But, the question arises how to construct triangles?

A unique triangle can be easily constructed using the concept of Geometry.

Geometry is a branch of Mathematics that deals with lines, angles, shapes, size, and dimension of different things we observe in everyday life. In Euclidean Geometry, there are different two dimensional and three -dimensional shapes. Flat shapes such as square, triangle, and circle are known as two -dimensional shapes. These shapes have only length and width.

Solid shapes such as cube, cuboid, sphere, cone, etc are three-dimensional shapes. These shapes have length, width, or height.

These geometric shapes can be easily constructed using compass, ruler, and protractor. Let us learn the steps of constructing triangles with compass, ruler, and protector below.

To construct a triangle when hypotenuse and one side is given, we need the following geometric tools:

A Ruler

A Compass

Let us learn to construct a triangle when hypotenuse and one side is given through examples:

Construct a right-angled triangle ABC with the length of the hypotenuse AB = 3 cm and side BC = 5 cm. The steps of construction are:

Step 1:

Draw a line of any length and mark a point C on it.

Step 2:

Set the width of the compass to 3 cm.

Step 3:

Place the pointer of the compass at C and draw an arc on both sides of C.

Step 4:

Mark the point as P and A where both the arc crosses the line.

Step 5:

Taking P as the centre, draw an arc above the point C.

Step 6:

Taking A as the centre, draw an arc that cuts the previous arc.

Step 7:

Mark the point B, where two arcs intersect each other.

Step 8:

Join the points B and A along with B and C with the help of the ruler.

Thus, ΔABC is the required right-angled triangle.

To construct a triangle when two sides and angle, we need the following geometric tools:

A Ruler

A Protractor

A Compass

Let us learn to construct a triangle when the length of two sides and included angle are given through an example.

Example

Construct a triangle PQR with PQ = 4 cm, QR = 6.5 cm , and ∠PQR = 60°.

The steps of construction are:

Step 1:

Draw a line QR = 6.5 cm using a ruler.

Step 2:

Using protractor at Q, draw a line QX making an angle of 60° with QR

Step 3:

Taking Q as the centre, draw an arc of radius 4 cm to cut the line QX at P.

Step 4:

Join PR.

Therefore, PQR is the required triangle.

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To construct a triangle when two sides and angle, we need the following geometric tools:

A Ruler

A Protractor

Let us learn to construct a triangle when the length of one side and included angle are given through examples.

Construct a triangle XYZ with XY = 4 cm, ∠ZXY = 100° and ∠ZXY = 30°.

The Steps of Construction are:

Step 1:

Draw a line segment XY = 4 cm using a ruler

Step 2:

Using protractor at X, draw a ray XP forming an angle of 30° with XY

Step 3:

Using protractor at Y, draw another ray YQ making an angle of 100° with XY

Step 4:

Let the rays XP and QY intersect at Z.

Step 5:

Using the property, sum of all the angles of a triangle is equal to 180°, we can easily find the third angle of the triangle which is 50°. Hence, ∠Z = 50°.

Step 6:

Hence, XYZ is a required triangle.

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To construct a triangle when all the three sides are given, we need the following geometric tools:

A Ruler

A Protractor

Before knowing how to construct a triangle with given sides, we should check the following property of triangles is met by the length of all the three sides.

“ The sum of all the three sides of a triangle should always be greater than its third side”.

We will not be able to construct a triangle with the given three sides if the above-mentioned property is not met by the given three sides.

Let us learn to construct a triangle given three sides through an example.

Example:

Construct a triangle ABC with side AB = 4 cm, BC = 6 cm and AC = 5 cm.

The steps of construction are:

Step 1:

Draw a line BC = 6 cm ( the longest side).

Step 2:

Taking B as centre, draw an arc of radius 4 cm above the line segment BC.

Step 3:

Taking C as centre, draw an arc of radius 5 cm that intersects the previous arc at ‘A’.

Step 4:

Join line segments AB and AC

Hence, ABC is the required triangle.

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Construct an isosceles triangle PQR with PQ = 6 cm, QR = 6 cm and ∠PQR = 50°.

Steps of drawing a triangle with protractor for the given sides and angles are as follows:

Draw a line QR 6 cm long.

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Taking Q as the centre, draw an angle of 50° using the protractor.

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Taking R as the centre, draw an angle of 50° using the protractor (angles opposite to the equal sides of an isosceles triangle are equal).

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Mark the point P where two lines intersect.

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Therefore, PQR is the required isosceles triangle.

1. Construct an equilateral triangle with a side 5 cm long using a protractor?

Ans: An equilateral triangle is a triangle whose all the three sides are equal in length. Another property of the equilateral triangle is that three angles of the triangle are equal, and each angle of a triangle is equal to 60 degrees.

Following are steps to construct an equilateral triangle with each side 5 cm long.

Step 1:

Draw a line AB of 5 cm long.

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Step 2:

Taking A as centre, draw an angle of 60° using a protractor.

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Step 3:

Taking B as centre, draw another angle of 60° using a protractor.

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Step 4:

Mark the point C where both the lines meet.

Hence, ABC is a required equilateral triangle of length 5 cm.

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2. Write down the steps in constructing a triangle ABC with sides AB = 3.5 units, BC = 6 units and AC = 4.5 units.

Solution

Step 1:

Draw a line segment BC measuring 6 units.

Step 2:

With B as center, and draw an arc of radius 3.5 units

Step 3:

With C as center, draw an arc of radius 4.5 units to intersect the previous arc at A

Step 4:

Join the line segment AB and AC.

Hence, the triangle ABC is drawn.

Triangle is a polygon with the minimum possible number of sides (three).

Hatch marks, also known as tick marks are used in triangles to identify the sides of equal length.

Two triangles are considered similar if each angle of one triangle has the same measure as the corresponding angle in the other triangle.

FAQ (Frequently Asked Questions)

1. What are the Different Properties of Triangles?

Ans: The different properties of triangle are discussed below:

The sum of all the interior angles of a triangle is equal to 180 degree. This is known as the angle sum property.

The side opposite to the largest angle is the longest side of a triangle whereas the side opposite to the smallest angle is the smallest side of the triangle.

The exterior angle property of a triangle states that the exterior angle of a triangle is equal to the sum of its interior opposite angles.

The sum of the length of any two sides of a triangle is greater than its third side whereas the difference of the length of any two sides of a triangle is less than its third side.

2. What are the Conditions that Determine a Unique Triangle?

Ans: Following are the conditions that determine a unique triangle:

Three sides of a triangle.

Two sides of a triangle and included angle.

Two angles of a triangle and included sides.

Two angles and the side opposite to the given angle of a triangle.

3. What are the Different Types of Triangles?

Ans: Based on the angles and side length, triangles are categorized into 6 six different types. Based on the side length, triangles are classified as:

Scalene Triangle

Isosceles Triangle

Equilateral Triangle

Based on the angles, triangles are classified as:

Acute Angled Triangle

Obtuse Angled Triangle

Right Angled Triangle