Isosceles Obtuse Triangle

An isosceles obtuse triangle is a triangle in which one of the three angles is obtuse (lies between 90 degrees and 180 degrees) and the other two acute angles are equal in measurement. One example of isosceles obtuse triangle angles is 30°, 30°, and 120°.

In geometry, an isosceles obtuse triangle can be considered as the triangle that contains the properties of both an isosceles triangle and an obtuse triangle. Let us recap the meaning of isosceles triangles and obtuse triangles.

• An isosceles triangle is one in which any two angles of the triangle are equal in measurement. And the two sides opposite to those equal angles are also equal in length.
• An obtuse triangle is one in which one of the angles lies between 90 degrees and 180 degrees and the other two angles are acute (less than 90°).

Look at the picture of an obtuse isosceles triangle given below to understand how it appears. △ABC is an example of an isosceles obtuse triangle with an obtuse angle of 120° at vertex A and two equal acute angles at vertices B and C. The sides opposite to the equal angles (AB and AC) are equal in length.

It is easy to identify an obtuse isosceles triangle if we know its properties. The properties of the isosceles obtuse triangle are listed below:

• It contains two acute angles and two sides opposite to those angles are equal.
• One of the angles is between 90° and 180° and the other two angles are acute angles each less than 45 degrees.
• The largest side of an obtuse isosceles triangle is the side opposite to the obtuse angle.
• The side opposite to the obtuse angle is the largest side. In other words, in an isosceles obtuse triangle, the unequal side is the largest.
• The sum of all the interior angles is 180 degrees.

The formula of an isosceles obtuse triangle is useful to find the area and perimeter of the triangle. There are two possible formulae that can be used to find the area of an isosceles obtuse triangle based on what information is given to us.

• If the length of base and height of the triangle is given, then area = [1/2 × base × height] square units.
• If the length of all three sides are given, then area = \((s-a) \sqrt{s(s-b)}\) square units, where, s = perimeter/2 = (2a + b)/2, a is the length of the equal sides, and b is the length of the unequal side. This formula is derived by using Heron's formula. Let's see how.

Applying Heron's formula to find the area of a triangle, we have, \(\sqrt{(s)(s-a)(s-b)(s-c)}\), where s is the semi-perimeter and a, b, and c are the sides of the triangle. But in the case of an isosceles obtuse triangle, two of the sides are equal. So let us assume a = c. By substituting the value of 'c' in the above formula, we get, \(\sqrt{(s)(s-a)(s-a)(s-b)}\).

⇒ \(\sqrt{(s)(s-a)^{2} (s-b)}\)

⇒ \((s-a)\sqrt{(s)(s-b)}\)

where a and b are the sides of the triangle and s is the semi-perimeter, which is (a + a + b)/2 or (2a+b)/2. Look at the image given below showing isosceles obtuse triangle formulas for finding area and perimeter.

To find the isosceles obtuse triangle perimeter, we just have to add the length of all three sides. So, the perimeter of an isosceles obtuse triangle = (2a + b) units, where a is the length of the equal side and b is the length of the unequal side of the triangle.

In an obtuse isosceles triangle, there is only one line of symmetry. It divides the triangle into two equal parts. The line of symmetry in an obtuse isosceles triangle divides a triangle into two equal areas such that if we fold it along the line, we will get two exact copies of the triangle. In the image given below, the line of symmetry divides the triangle ABC into two equal parts. The angle A of 120° is divided into two angles of 60 degrees each with the line of symmetry.

► Also check:

• Types of Triangles
• Obtuse Triangle
• Isosceles Triangle
• Right Scalene Triangle
• Obtuse Scalene Triangle

What is an Isosceles Obtuse Triangle?

An isosceles obtuse triangle is a triangle that comes in the category of both obtuse triangles and isosceles triangles. In this triangle, there is one obtuse angle and the other two angles are equal in measurement and are acute angles. One example of the obtuse isosceles triangle angles is 40°, 40°, and 100°.

Is an Obtuse Isosceles Triangle Possible?

Yes, it is possible to draw an isosceles obtuse triangle. We just need one obtuse angle and two acute angles each less than 45° and equal in measurement. Some examples of isosceles obtuse triangle angles are given below:

• 30°, 30°, and 120°
• 25°, 25°, and 130°
• 40°, 40°, and 100°
• 15°, 15°, and 150°

How to Find the Area of an Isosceles Obtuse Triangle?

The area of an obtuse isosceles triangle can be calculated by using the formula: Area = 1/2 × base × height square units. The base is the side opposite to the vertex from where the height is drawn or measured.

What are the Properties of an Isosceles Obtuse Triangle?

The properties of an isosceles obtuse triangle are listed below:

1. It has one obtuse angle and two acute angles which are equal in measure and each less than 45 degrees.
2. Have two equal sides and two equal angles.
3. The two sides opposite to the equal angles are equal.
4. The side opposite to the obtuse angle is the largest side of an isosceles obtuse triangle.
5. The sum of all the interior angles should be 180 degrees.
6. The sum of two equal acute angles is always less than 90 degrees.

How do you Draw an Isosceles Obtuse Triangle?

To draw an isosceles obtuse triangle, the easiest way is to first draw a line segment horizontally which will be the base of the triangle. Now, draw two angles of equal measurements (each should be less than 45 degrees) on both the ends of the line segment. Join the arms of the angles. In this way, we will get an obtuse isosceles triangle.

What does an Isosceles Obtuse Triangle Look Like?

An isosceles obtuse triangle looks like an obtuse triangle with two equal sides and two equal angles each less than 45 degrees.