What is an Obtuse Triangle?
An obtuse angled triangle, or just an obtuse triangle, is a triangle in which any one of the angles is more than 90 degrees. As a result, if one angle is obtuse or greater than 90 degrees, the other two angles must be acute or less than 90 degrees.
Below is an example of an obtuse triangle, △ABC, in which ∠B is obtuse and other two angles are acute.
Since one angle is obtuse in an obtuse triangle so the sum of the other two angles is less than 90°.
[Note: The side opposite the obtuse angle is the longest side of the obtuse triangle.]
Types of Obtuse Triangles
There are two types of obtuse triangles:
- Obtuse Scalene Triangle : An obtuse triangle with all three sides of different lengths.
2. Obtuse Isosceles Triangle: An obtuse triangle with any two sides equal in length.
Properties of Obtuse Triangles
- An obtuse triangle has one obtuse angle and two acute angles.
- The longest side of an obtuse triangle is the side that is opposite to the obtuse angle.
- There can only be one obtuse angle in a triangle.
- In an obtuse triangle, the sum of the two acute angles is less than 90°.
Formula for the Perimeter of an Obtuse Triangle
The perimeter of an obtuse triangle can be measured using the formula:
Perimeter, P = a + b + c
Where,
‘a’, ‘b’ and ‘c’ denote the side lengths of the triangle. Perimeter is measured in the same units as the side length.
Formula for the Area of an Obtuse Triangle
The area of an obtuse triangle can be measured using the formula:
Area, \( A=\left ( \frac{1}{2} \right )\times~b\times~h\) square units.
Where,
‘b’ denotes the base and ‘h’ is the height of the triangle.
Solved Examples
Example 1: Find the missing angle in the following obtuse triangle.
Solution:
Assume ‘x’ as the missing angle.
\( 180^\circ~=~x~+~24^\circ~+~40^\circ\) [Angle sum property of triangle]
\( 180^\circ~=~x~+~64^\circ\) [Add]
\( 180^\circ~-~~24^\circ~=~x\) [Subtract \( 64^\circ\) on each side]
\( 116^\circ~=~x\) [Subtract]
Therefore, the missing angle in the given obtuse triangle is 116 degrees.
Example 2: The area of an obtuse angled triangle is 25 square centimeters. Find its height if its base measures 5 centimeters.
Solution:
Apply the formula for the area of a triangle:
Area, \( A~=~\frac{1}{2}~\times~base~\times~height\)
\( 25~=~\frac{1}{2}~\times~5~\times~height\) [Substitute the given values]
\( 25\times~2~=~5~\times~\times~height\) [Multiply both sides by 2]
\( 50~=~5~\times~height\) [Simplify]
\( 10~=~height\) [Divide both sides by 5]
So, height= 10 cm
Hence, the height of the given obtuse triangle is 10 centimeters.
Example 3: Identify obtuse angled triangles among the following:
Solution:
Observe the measure of each interior angle of the given four triangles.
For triangle B one of the angles is \(118^\circ\) and for triangle C one of the angles is \(104^\circ\).
So triangles B and C each have one angle greater than 90 degrees.
Therefore, among the given triangles, triangle B and triangle C are obtuse angled triangles.
A triangle cannot have a right angle and an obtuse angle as it would not satisfy the angle sum property of triangles.
The measure of interior angles of a triangle is an indication of whether a triangle is obtuse or not. If one of the angles is greater than 90 degrees, then, the triangle is an obtuse angled triangle.
The sum of interior angles of a triangle is 180 degrees, and so a triangle cannot have two obtuse angles.