Obtuse Angled Triangle

A triangle is a closed two-dimensional plane figure with three sides and three angles. Based on the sides and the interior angles of a triangle, different types of triangles are obtained and the obtuse-angled triangle is one among them. If one of the interior angles of the triangle is obtuse (i.e more than 90°), then the triangle is called the obtuse angle triangle. The obtuse angle in the triangle can be any one of the three angles and the remaining two angles are acute angles.

The obtuse angle triangle properties are different from other triangles. Look at the table below for the same:

Triangle type Obtuse Acute Right
Difference Any One angle more than 90° All angles less than 90° One angle equal to 90°

Obtuse Angled Triangle Formula

Area = \(\frac{1}{2}\times b\times h\)

Or

A=\(\sqrt{S(S-a)(S-b)(S-c)}\)sq.units

Where S = \(\frac{a+b+c}{2}\) (s= semiperimeter)

Perimeter = a+b+c

How do you know if a triangle is obtuse?

If any two angles of a triangle are given, it can be easily determined whether the triangle is an obtuse triangle or not. But how to determine this, when the three sides of the triangle are known? We have an inequality in the lines of Pythagorean identity to test this.

The triangle is an obtuse triangle if the sum of the squares of the smaller sides is less than the square of the largest side.

Let a, b and c are the lengths of the sides of triangle ABC and c is the largest side, then the triangle is obtuse if

a2 + b2 < c2

Obtuse Angled Triangle Properties

  1. The sum of the two angles other than the obtuse angle is less than 90 degrees.
  2. The side opposite to the obtuse angle is the longest side of the triangle.
  3. An obtuse triangle will have one and only one obtuse angle. The other two angles are acute angles.
  4. The points of concurrency, the Circumcenter and the Orthocenter lie outside of an obtuse triangle, while Centroid and Incenter lie inside the triangle.

Example

Triangle ABC is a perfect example to study the triangle type – Obtuse.

  • In triangle ABC, interior angle ACB =37°, which is less than 90°, so it’s an acute angle.
  • Interior angle ABC = 96°, which is more than 90° so, it’s an obtuse angle.
  • Interior angle BAC=47°, which is less than 90°, so it’s an acute angle.
  • As this triangle, ABC has one angle (ABC=96 degree) more than 90°, so this triangle is obtuse.