Isosceles Triangle

Definition of Isosceles Triangle: An isosceles triangle is a triangle having at least two sides of equal length. The term isosceles is derived from the Greek words isos meaning equal and skelos meaning leg, emphasizing that two of the triangle's sides are of the same length, while the third side may be of a different length. The angles opposite the equal sides are also equal.

A common real-world example of an isosceles triangle is the shape of many rooftops, where two sides of the roof are equal in length, forming the "legs" of the triangle, and the peak of the roof forms the vertex.

Properties of an Isosceles Triangle

• Legs

  : The defining characteristic of an isosceles triangle is that it has two sides of equal length. These sides are often referred to as the legs of the triangle.

• Base

  : The third side of the triangle, which may not equal in length to the other two, is called the base of the triangle.

• Base Angles

  : The angles opposite the two equal sides are also equal in measure. These angles are referred to as the base angles.

• Base Angle Theorem

  : If two angles of a triangle are equal in measure, then the sides opposite those angles are also equal. In an isosceles triangle, this means that the base sides are equal in length.

• Height or Altitude

  : The height or altitude of an isosceles triangle is the perpendicular line drawn from the vertex (the top point where two equal sides intersect) to the base. It bisects the base and creates two congruent right triangles.

Types of an Isosceles Triangle

Acute Isosceles Triangle

In an acute isosceles triangle, all three angles are less than 90 degrees. The two equal sides are typically referred to as the "legs," while the base is the side that is not equal in length.

Right Isosceles Triangle

A right isosceles triangle has one angle that measures exactly 90 degrees, making it a right triangle. The two equal sides are adjacent to the right angle, and the hypotenuse is the longest side opposite the right angle. The angles in a right isosceles triangle are 90 degrees, 45 degrees, and 45 degrees.

Obtuse Isosceles Triangle

An obtuse isosceles triangle has one angle that is greater than 90 degrees. This means that the two equal sides enclose an angle that exceeds a right angle, while the remaining angle is less than 90 degrees. The sum of all three angles still equals 180 degrees.

Isosceles Triangle Formulas

Perimeter of Isosceles Triangle

To calculate the perimeter of a triangle, you need to add the lengths of all three sides together. In an isosceles triangle, there are two sides (the legs) that are of equal length ($a$) and one side (the base) that may be of a different length ($b$). Simply add twice the length of one of the equal sides ($2a$) to the length of the base ($b$) to find the perimeter of the isosceles triangle.

Area of Isosceles Triangle

1. Given the base (b) and height (h)

   $\mathrm{Area}={\displaystyle \frac{1}{2}}\times b\times h$

   where

   • $b$ = length of the base (the unequal side).
   • $h$ = height (perpendicular distance from the base to the opposite vertex)

2. Given all three sides ($a$, $a$, $b$) (using Heron’s Formula):

   $\mathrm{Area}=\sqrt{s(s-a)(s-a)(s-b)}$

   where

   • $s$ is semi-perimeter, $s=\frac{2a+b}{2}$
   • $a$ = length of one of the two equal sides.
   • $b$ = length of the base (the unequal side).

3. Given all three sides ($a$, $a$, $b$) and no height:

   $$\mathrm{Area}=\frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}$$

   where

   • $b$ = length of the base (the unequal side).
   • $a$ = length of one of the two equal sides.

4. Given the lengths of two equal sides ($a$) and the included angle ($\theta$)

   $$\mathrm{Area}=\frac{1}{2}\times a^2\times \sin \theta$$

   where

   • $a$ = length of one of the two equal sides.
   • $\theta$ = angle between the two equal sides.

5. Given the lengths of equal sides ($a$) and base(unequal side) ($b$) and the included angle ($\theta$)

   $$\mathrm{Area}=\frac{1}{2}\times a\times b\times \sin \theta$$

   where

   • $a$ = length of one of the two equal sides.
   • $b$ = length of the base (the unequal side).
   • $\theta$ = angle between one of the two equal sides and unequal side.

6. Given the measures of base angles($\alpha$), apex angle($\beta$) and length between them(equal side, $a$)

   $$A=\frac{a^2\times sin(\beta )\times sin(\alpha )}{2\times sin(2\pi -\alpha -\beta )}$$

   where

   • $\alpha$ = base angle (equal angles opposite the equal sides)
   • $\beta$ = apex angle (angle opposite the triangle's base)
   • $a$ = length of one of the two equal sides (side between $\alpha$ and $\beta$)

7. Given the lengths of two equal sides ($a$) of isosceles right triangle

   $$\mathrm{Area}=\frac{1}{2}\times a^2$$

   where $a$ = length of one of the two equal sides.

Altitude of Isosceles Triangle

Solved Examples on Isosceles Triangle