Obtuse Isosceles Triangle (Definition, Example, Formula, Height, Perimeter, Area, Properties)

Obtuse Isosceles Triangle Definition

The term Obtuse Isosceles Triangle is made up of three words i.e., “Obtuse“, “Isosceles” and “Triangle“. In mathematics, the word “Obtuse” refers to the angles that are greater than 90^0 and the word “Isosceles” refers to the triangles in which any two sides are of equal length. Therefore, The Obtuse Isosceles Triangle may be defined as the triangle in which one angle is greater than 90^0 and two sides that forms the Obtuse Angle are equal in length.

Obtuse Isosceles Triangle Example

The example of an Obtuse Isosceles Triangle is shown in the figure below.

Obtuse Isosceles Triangle

From the Obtuse Isosceles Triangle \vartriangle ABC as shown in the figure above,

  • The angle \angle ABC is greater than 90^0.

\angle ABC >90^0

  • The sides \overline {AB} and \overline {BC} are equal to each other.

\overline {AB}=\overline {BC} and

  • Since the two sides \overline {AB} an \overline {BC} of the Obtuse Isosceles Triangle \vartriangle ABC is equal, the two angles made by the sides with the third side is also equal.

\angle BAC=\angle BCA

Height of an Obtuse Isosceles Triangle

Obtuse Isosceles Triangle Height

As shown in the figure above, the two equal sides of the Obtuse Isosceles Triangle \vartriangle ABC is denoted by the letter a and the base is denoted by the letter b. The Height (h) of the given Obtuse Isosceles Triangle can be found out by using the formula:

h=\frac{1}{2}\sqrt{4a^2-b^2}

Perimeter of Obtuse Isosceles Triangle

The Perimeter of Obtuse Isosceles Triangle may be defined as the sum of the length of all three sides of the triangle. If the two equal sides of the triangle are denoted by the letter a and the third side is denoted by b, then the Perimeter of Obtuse Isosceles Triangle will be,

P=a+a+b

P=2a+b

Semiperimeter of Obtuse Isosceles Triangle

The Semiperimeter of Obtuse Isosceles Triangle is the half of the Perimeter of the triangle and the same is given by,

s=\frac{2a+b}{2}

Area of Obtuse Isosceles Triangle

The area of an Obtuse Isosceles Triangle can be found out by using the following methods:

Area of Obtuse Isosceles Triangle using Base and Height

If the Base(b) and the Height(h) of an Obtuse Isosceles Triangle is known, then the area of the triangle can be found out by using the formula:

A=\frac{1}{2}\times b\times h

Area of Obtuse Isosceles Triangle using Apex Angle

The Apex Angle of an Obtuse Isosceles Triangle is the angle made by the two equal sides with each other. As shown in the figure below, the Apex Angle is \theta.

Now, if the Apex Angle(\theta) and the two equal sides a of the Obtuse Isosceles Triangle are known, then the area of the triangle can be found out by using the formula,

A=\frac{1}{2}a^2 sin\theta

Area of Obtuse Isosceles Triangle using Heron’s Formula

If only the three sides of an Obtuse Isosceles Triangle are known, then the area of the triangle can easily be found out by using Heron’s Formula and the same is given by,

A=\sqrt{s(s-a)(s-a)(s-c)}

A=\sqrt{s(s-a)^2(s-c)}

A=(s-a)\sqrt{s(s-c)}

Obtuse Isosceles Triangle Properties

The Properties of an Obtuse Isosceles Triangle are as follows:

  1. One out of the three angles in an Obtuse Isosceles Triangle is always greater than 90^0.
  2. The two angles other than the Obtuse Angle in an Obtuse Isosceles Triangle is of equal measure.
  3. The two sides that make the Obtuse Angle in an Obtuse Isosceles Triangle are of equal length.

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Frequently Asked Questions (FAQ)

The Obtuse Isosceles Triangle may be defined as the triangle in which one angle is greater than 90º as well as any two sides of the triangle are equal in length

The Area of an Obtuse Isosceles Triangle can be found out by using three methods as given below:
1. If the Height and the Base of the Triangle is known: A=(1/2)bh
2. If the Apex Angle of the Triangle is known: A=1/2 a² sinθ
3. If the length of the three sides of the triangle is known: A=(s-a)√{s(s-c)}, where s is the semi perimeter and can be found out by using the formula, s=(2a+b)/2.

Yes, an Isosceles Triangle can be an Obtuse Triangle. This kind of triangles are known as Obtuse Isosceles Triangle.

An Isosceles Triangle can have only one Obtuse Angle.

The Obtuse Isosceles Triangle look like the image given below. One angle of the triangle is the obtuse angle and the sides that form the obtuse angle are equal in length.

Obtuse Isosceles Triangle

Yes, the Obtuse angled Isosceles Triangle will look like
Obtuse Isosceles Triangle

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