Triangles

Table of Contents 


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Introduction to Triangle

A closed figure made with 3 line segments forms the shape of a triangle.

Triangles are an important part of our lives.

You can observe the shape of a triangle in signboards, your favourite sandwiches or birthday banners.

Sandwich which is triangular in shape

Birthday banners are examples of a triangle shape

Triangle shape example of a sign board


What is a Triangle?

A triangle is a closed figure or shape with 3 sides, 3 angles and 3 vertices.

A triangle with vertices P, Q, and R is denoted as △PQR.

The elements of a triangle are its 3 angles and its 3 sides.

Let's look at triangle PQR.

 A triangle PQR

The elements of this triangle are:

  • \(\angle PQR\)
  • \(\angle QRP\)
  • \(\angle RPQ\)
  • PQ
  • QR
  • RP

You can explore the relation between the elements of a triangle by dragging the vertices of the triangle shown below.


Different Types of Triangles

Triangles can be classified based on their angles or the length of their sides.

Classification of Triangles by Angles

Type of Triangle  Example Explanation
Right-angled triangle A right-angled triangle ACB

A right triangle has one of its vertex angles as 90°.

In the above triangle, \(\angle \text{ACB}\) = 90°.

Acute-angled triangle Acute-angled triangle DEF - all angles are less than 90 degrees

An acute triangle has all its internal angles as acute( i.e.. less than 90°).

Obtuse-angled triangle  Obtuse-angled triangle DFE

An obtuse triangle has one of its internal angles as obtuse ( i.e.. greater than 90°). 

In the given triangle, \(\angle \text{DFE}\)= 120\(^\circ\)

Classification of Triangles by Sides

Type of Triangle  Example Explanation
Equilateral triangle Equilateral triangle ABC has all sides equal

If all the three sides of a triangle are equal, it is called an equilateral triangle.

In the given triangle, all three sides are equal in length.

AB = BC = AC

 

Isosceles triangle  Isosceles triangle RST has two sides equal.

If any two sides of a triangle are equal, it is called an isosceles triangle.

In the given triangle, two sides are equal.

RS = RT

 

Scalene triangle  A scalene triangle XYZ has all sides unequal.

If a triangle has three unequal sides, it is called a scalene triangle.

In the given triangle, all three sides are of unequal lengths.

XZ ≠ XZ ≠ YZ

 


Fun Facts on Triangles

  • A triangle can always be divided into two right triangles, irrespective of its orientation.
  • Triangles are polygons with the least number of sides. There are no polygons with less than 3 sides.
  • Many towers, buildings and monuments are constructed in the shape of triangles to give them more stability and strength.
    Here are few examples for you:

Eiffel tower is made up of many triangles

1. Eiffel Tower

The world-famous Eiffel Tower is triangular in shape and has around 186 triangles in its structure

Egyptian pyramids have faces as triangles

2. Egyptian Pyramids

They have triangular faces

Real life example - triangles are used in bridge frames for stability

3. Bridges

They use triangular frames for stability


Properties of a Triangle

Some of the important properties of a triangle are listed below.

Property 1 - Angle Sum Property

The Angle Sum Property states that the sum of the three interior angles of a triangle is 180°

Acute-angled triangle - all angles are less than 90 degrees.

In the given triangle,

\(\angle \text{D} + \angle \text{E} + \angle \text{F}  = 180^\circ \)

Property 2 - Triangle Inequality Property

The Triangle Inequality Property sum of the length of the two sides of a triangle is greater than the third side.

Triangle inequality property illustrated using two triangles

in \(\Delta PQR \)

  • p + q > r  ( 6 + 4 > 3)
  • r + p > q (3 + 4  > 6)
  • r + q > p ( 6 + 3 > 4)

Property 3 - Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.

\(\begin{align}\text{Hypotenuse}^2 &\!=\!\text{Base}^2 \!+\!\text{Altitude}^2
\end{align}\)

A right triangle with base, altitude and hypotenuse

Property 4 - Side opposite the greater angle is the longest side

To understand the side opposite the greater angle is the longest side property, let's consider the below-given triangle.

The side opposite to the greater angle is the longest side.

In this triangle, \(\angle \text{A}\) is the greatest angle.

Thus, side BC is the longest side.

Property 5 - Exterior Angle Property

The Exterior Angle Property of a triangle is always equal to the sum of the interior opposite angles. 

Exterior angle of a triangle is the sum of interior opposite angles.

In the given triangle, Exterior angle E1  = \(\angle \text{ABC} + \angle \text{BCA}\)

A triangle has 3 exterior angles and these exterior angles add up to 360° for any polygon.

Property 6 - Congruence Property

In the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are congruent.

Two congruent triangles have all its elements measuring the same.

All  the corresponding angles and sides are equal,

  • \(\angle\text{ ABC} = \angle\text{PQR}\)
  • \(\angle\text{ BAC} = \angle\text{QPR}\)
  • \(\angle\text{ BCA} = \angle\text{QRP}\)
  • AB = PQ
  • AC = PR
  • BC = QR 

Formulas of a Triangle

The formulas of a triangle help us determine its area and perimeter.

Perimeter of a Triangle

Consider △XYZ.

The length of the sides of the triangle are represented as ‘a’, ‘b’ and ‘c’ respectively.

 Triangles with sides a,b and c; perimeter is a plus b plus c

\(\text{Perimeter of Triangle} = a + b + c\)

Area of a Triangle

Area of triangle formula is shown using a triangle with height h.

In the given triangle \(ABC\),

\(\begin{align}\text{Area of }\Delta ABC  = \frac{1}{2} h\times \text{BC}\end{align}\)

\(\text{BC}\) is the base and \(h\) is the height of the triangle.

\(\begin{align}\frac{1}{2} \times \text{Base} \times \text{Height}\end{align}\)
 
tips and tricks
Tips and Tricks
  1. A triangle cannot have more than one right angle.
  2. Each angle of a triangle cannot be less than 60°.
  3. The angles opposite to the equal sides of an isosceles triangle are equal.

Help your child score higher with Cuemath’s proprietary FREE Diagnostic Test. Get access to detailed reports, customised learning plans and a FREE counselling session. Attempt the test now.

Solved Examples

Example 1

 

 

The perimeter of a triangular garden is given as 13 m.

If two of its sides measure 3 m and 4 m respectively, what is the measure of the third side?

 A triangular shaped garden

Solution:

We know that the perimeter of a triangle is the sum of all three sides

\(\implies\) 13 = 3 + 4 + unknown side

Therefore, the unknown side is given by:

\(\begin{align} &=13 \!-\! (3 \!+\! 4) \\ &= 6 \;m \end{align}\)

\(\therefore\)The third side of the park measures 6 m
Example 2

 

 

Tina is building a triangular, wooden birdhouse as shown.

If two of the angles measure 46° and 62°, what is the measure of the third angle?

 A triangular bird house; two angles are given, find the third angle

Solution:

We know that the sum of the angles of a triangle add up to 180°.

Therefore, the unknown angle is:

\(\begin{align} &=180^\circ \!-\! (46^\circ \!+\! 62^\circ) \\ &= 72^\circ\end{align}\)

\(\therefore\)The third angle measures 72°
Example 3

 

 

In a right triangle, two of the sides measure 3 cm and 4 cm.

What is the measure of the hypotenuse?

 Find the hypoteneuse of a right triangle whose other two legs measure 3 cm and 4 cm

Solution:

By Pythagoras theorem, we know that, hypotenuse2 = base2 + height2

Substituting the values, we get:

\(\begin{align}
\text{hypoteneuse}^2 &=3^2 +4^2 \\
&= 9 + 16\\
&= 25\: \text{cm} \\
\implies \text{hypoteneuse} &= 5 \:\text{cm}
\end{align}\)

\(\therefore\) Hypotenuse = 5 cm
Example 4

 

 

One of the acute angles of a right-angled triangle is 45°

Find the other triangle.

Identify the type of triangle thus formed.

A right triangle and one of the other angle is 45 degrees. Find the other.

Solution:

Given, angle1 =  90° and angle2 = 45°

We know that the sum of the angles of a triangle add up to 180°.

Therefore,

\(\begin{align} \text{angle}_3 &=180^\circ \!-\! (90^\circ \!+\! 45^\circ) \\ &= 45^\circ\end{align}\)

Since two angles measure the same, it is an isosceles triangle.

\(\therefore\) △ABC is an isosceles triangle
Example 5

 

 

The sides of a triangular park are in the ratio 3: 4: 5 and its perimeter is 1080 m.

Find the area of the triangular park.

 3 sides of a triangle are in the ratio 3 is to 4 is to 5 and the perimeter is 1080 m. Find the area of the park.

Solution:

Given:

  • Perimeter = 1080 m
  • Sides of the triangle are in the ratio 3:4:5.

Therefore, the sides of triangle are

\( \begin{align}
\text{side}_1 &= \frac{3}{12}\times 1080\\\ &= 270 \text{m}\\\\
\text{side}_2 &= \frac{4}{12}\times 1080\\\ &= 360\text{m} \\\\
\text{side}_3 &= \frac{5}{12}\times 1080\\\ &= 450\text{m}
\end{align} \)


\(\begin{align} \text{Area of the park} &=    \frac{1}{2} \times 270 \times 360 \\
&= 48600\: \text{m}^2\end{align} \)

\(\therefore\)Area of the park = 48,600 m2
 
important notes to remember
Important Notes
  1. A triangle is a 3-sided closed shape.
  2. The perimeter of a triangle is the sum of all three sides.
  3. The area of a triangle is half the product of its height and base.
  4. Herons formula and Pythagoras theorem are important formulas related to triangles.
  5. The sum of the angles of a triangle add up to 180°.

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Practice Questions

Here are few questions for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Area of a triangle worksheets are available at the end of the page for you to try and solve!

 
Challenge your math skills
Challenging Questions

     

1. Anu said a triangle has at least one set of parallel lines. Is she right?  Why? 
  can a triangle have parallel lines?
2. Can a triangle be formed with dimensions 17cm, 9 cm and 30cm? 
        

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions(FAQs)

 1. What is a triangle?

A three-sided closed shape made with 3 line segments is called a triangle.

2. What are the types of triangles?

  • Types of triangles based on sides:
    • Scalene
    • Isosceles
    • Equilateral
  • Types of triangles based on angles:
    • Acute triangle
    • Obtuse triangle
    • Right triangle

3. How many properties does a triangle have?

The 6 important properties of triangles are listed in section Properties of a Triangle.

 
Download Triangles and Quadrilaterals Worksheets
Triangles and Quadrilaterals
grade 9 | Questions Set 1
Triangles and Quadrilaterals
grade 9 | Answers Set 1
Triangles and Quadrilaterals
grade 9 | Answers Set 2
  
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