Obtuse Triangles

Obtuse Triangles

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Introduction to Triangle

We know that triangles are 3-sided closed shapes made with 3 line segments.

Explore the different triangles and their elements visually using the simulation below.


Definition of Obtuse Triangle

An obtuse triangle is a type of triangle where one of the vertex angles is greater than 90°

Examples of obtuse triangles

Examples of obtuse triangles

The triangles above have one angle greater than 90°.

Hence, they are called obtuse-angled triangle or simply obtuse triangle.

An obtuse-angled triangle can be scalene or isosceles, but never equilateral.


Obtuse Angled Triangle Formula

Perimeter

The perimeter of an obtuse triangle is the sum of the measures of all its sides.
 

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

Perimeter of obtuse triangle = a + b + c

Area

In the given obtuse triangle \(ABC\),

Area of an obtuse triangle

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

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

Area = \(\begin{align}\frac{1}{2} \times \text{Base} \times \text{Height}\end{align}\)

We know that a triangle has 3 altitudes from the 3 vertices to the corresponding opposite sides.

The altitude or the height from the acute angles of an obtuse triangle lie outside the triangle.

We extend the base as shown and determine the height of the obtuse triangle.

We can also find the area of an obtuse triangle area using Heron's formula. 

Area of triangle with 3 sides a, b, and c

Consider the triangle \(ABC\) with sides \(a\), \(b\) and \(c\).

Heron's formula to find the area of a triangle is:

\(\sqrt {s(s - a)(s - b)(s - c)}\)

Note that \((a + b + c)\) is the perimeter of the triangle.

\(s\) is the semi-perimeter which is given by:

\[s = \frac{{a + b + c}}{2}\]


Obtuse Angled Triangle Properties

Property 1

The longest side of a triangle is the side opposite to the obtuse angle.  

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

The side BC is the longest side which is opposite to the obtuse angle \(\angle \text{A}\)

Check the proof of this property here.

Property 2

A triangle cannot have more than one obtuse angle. 

Obtuse angle triangle property : only one angle in a triangle can be an obtuse angle.

We know that the angles of any triangle add up to 180°

Consider the obtuse triangle shown above.

We can observe that one of the angles measures greater than 90°, thus making it an obtuse angle.

Even if we assume this obtuse angle to be 91°, the other two angles of the triangle will add up to 89° degrees.

Hence, a triangle cannot have 2 obtuse angles.

Property 3

Sum of the other two angles in an obtuse-angled triangle is less than 90\(^\circ\)

We just learnt that when one of the angles is an obtuse angle, the other two angles add up to less than 90°

Obtuse angle triangle property : only one angle in a triangle can be an obtuse angle.

In the above triangle, 

\(\angle 1 >90^\circ \)

We know that by angle sum property, the sum of the angles of a triangle is 180°

Therefore, 

\(\angle 1 + \angle 2 +\angle 3 = 180 ^\circ  \) 

and

\(\angle 1 >90^\circ \)

Subtracting the above two, we have:

 \(\angle 2 +\angle 3 < 90 ^\circ \)

Property 4

The circumcenter and the orthocenter of an obtuse-angled triangle lie outside the triangle.

The orthocenter (H), the point at which all the altitudes of a triangle intersect, lies outside in an obtuse triangle. 

The orthocenter is the point at which all the altitudes of a triangle intersect.

Circumcenter (O), the point which is equidistant from all the vertices of a triangle, lies outside in an obtuse triangle. 

 Obtuse triangle property - circumcenter lies outside the triangle.

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Solved Examples

Example 1

 

 

Which of the following angle measures can form an obtuse triangle ABC?

a) 60°, 70°,50°

b) 95°, 30°, 55°

c) 89°,45°,46°

d) 90°, 60°, 30°

Example - Identify the possible angle measures for an obtuse triangle.

Solution:

An obtuse triangle has one of the vertex angles as an obtuse angle (> 90°).

Among the given options, option (b) satisfies the condition.

\(\therefore\) Option b forms an obtuse triangle.
Example 2

 

 

Find the area of an obtuse triangle whose base is 8 cm and height is 4 cm.

Area of an obtuse triangle of base 8 cm and height 4 cm.

Solution:

Area of an obtuse-angled triangle

= \(\begin{align}\frac{1}{2} \times \text{base} \times \text{height}\end{align}\)

Substituting the values of base and height, we get:

\(\begin {align}
\text{Area} &= \frac{1}{2} \times 8 \times 4 \\
&= 16\: \text{cm}^2
 \end{align}\)

\(\therefore\) Area of the triangle = 16 cm2
Example 3

 

 

Find the height of the given obtuse triangle whose area = 60 cm2 and base = 8 cm.

Area of an obtuse triangle is 60 square cm. Base is 8 cm.

Solution:

Area of an obtuse-angled triangle

= \(\begin{align}\frac{1}{2} \times \text{base} \times \text{height}\end{align}\)

Therefore, height of the obtuse triangle can be calculated by:

\(\begin{align}\text{Height} = \frac{2 \times \text{Area}}{\text{base}} \end{align}\)

Substituting the values, we get:

\(\begin {align}
\text{Height} &= \frac{2 \times 60}{8 }   \\
&= 15\: \text{cm}
 \end{align}\)

\(\therefore\) Height  = 15 cm
Example 4

 

 

Can sides measuring 3 cm, 4 cm and 6 cm form an obtuse triangle?

Solution:

The sides of an obtuse triangle should satisfy the condition that the sum of the squares of any 2 sides is greater than the third side.

We know that

  • \(a\) = 3 cm
  • \(b\) = 4 cm
  • \(c\) = 6 cm

Taking the squares of the sides, we get:

\(a^2\) = 9, \(b^2\) = 16 and \(c^2\) = 36

We know that 

36 > (9 +16 )

Therefore, the given measures can form the sides of an obtuse triangle.

3 cm, 4 cm and 6 cm can be the sides of an obtuse triangle.
Example 5

 

 

A triangle with one exterior angle measuring 80° is shown in the image.

Can this triangle be an obtuse triangle?

One of the exterior angle is 80 degrees. Is the triangle an obtuse triangle?

Solution:

The exterior angle and the adjacent interior angle forms a linear pair (i.e , they add up to 180°).

Hence, the interior angle at vertex B is 

180 - 80 = 100°

\(\therefore\) The given triangle is an obtuse-angled triangle.
 
tips and tricks
Tips and Tricks
  1. If the 3 sides of a triangle measure \(a\), \(b\) and \(c\) such that \(c\) is the longest side of the triangle, the triangle is an obtuse-angled triangle if \(a^2 + b^2 < c^2\)
  2. Centroid and incenter lie within the obtuse triangle while circumcenter and orthocenter lie outside the triangle.

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Practice Questions

Here are a few activities for you to practice.

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

 
 
 
 
 
 
 
important notes to remember
Important Notes
  1. An obtuse-angled triangle has one of its vertex angles as obtuse and other angles as acute angles.
  2. Perimeter of an obtuse triangle is the sum of the measure of all its sides and the area is \(\frac{1}{2} \times \text{base} \times \text{height}\).
  3. Circumcenter and orthocenter lie outside the triangle.

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 an obtuse triangle?

A triangle with one obtuse angle (greater than 90°) is called an obtuse triangle.

2. What is an example of an obtuse triangle?

A triangle where one angle is greater than 90° is an obtuse-angled triangle.

Example:  \(\Delta \text{ABC}\) has these angle measures \(\angle \text{A} = 120^\circ  , \angle \text{A} = 40^\circ , \angle \text{A} = 20^\circ \)

This triangle is an obtuse angled triangle because \( \angle \text{A} = 120^\circ\)

The following triangles are examples of obtuse triangles.

Examples of obtuse triangles

3. What are the properties of obtuse triangles?

  • The longest side of the triangle is the side opposite to the obtuse angle.
  • A triangle cannot have more than one obtuse angle.
  • Sum of the other two angles in an obtuse-angled triangle is less than 90\(^\circ\).
  • The circumcenter and the orthocenter of an obtuse-angled triangle lie outside the triangle.
  
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