How to Understand the Properties of Isosceles and Equilateral Triangles

Step-by-step Guide: Isosceles and Equilateral Triangles

1. Isosceles Triangle: Definition and Properties:
An isosceles triangle has two sides of equal length, known as the legs. The third side is called the base.

• The angles opposite the equal sides are congruent.
• The altitude drawn to the base bisects both the base and the vertex angle.

2. Equilateral Triangle: Definition and Properties:
An equilateral triangle has all three sides of equal length.

• All interior angles are congruent and each measure \(60^\circ\).
• All altitudes (or heights) are congruent.
• Every equilateral triangle is also equiangular.

3. Area and Perimeter Formulas:

For an isosceles triangle with base \(b\) and height \(h\):
\( \text{Area} = \frac{1}{2} b \times h \)
\( \text{Perimeter} = b + 2 \times \text{length of one leg} \)

For an equilateral triangle with side \(a\):
\( \text{Area} = \frac{\sqrt{3}}{4} \times a^2 \)
\( \text{Perimeter} = 3a \)

Examples

Example 1:
Given an isosceles triangle with a base of \(10 \text{ cm} \) and legs of \(12 \text{ cm} \) each, find its height.

Solution:
Using Pythagoras’ theorem for one half of the triangle (a right triangle):
\( h = \sqrt{\text{leg}^2 – \left(\frac{\text{base}}{2}\right)^2} \)
\( h = \sqrt{12^2 – 5^2} \)
\( h = \sqrt{119} \) or approximately \(10.9 \text{ cm} \)

Example 2:
Find the area of an equilateral triangle with a side length of \(8 \text{ cm} \).

Solution:
Using the area formula for an equilateral triangle:
\( \text{Area} = \frac{\sqrt{3}}{4} \times 8^2 = 55.4 \text{ cm}^2 \)

Practice Questions:

1. Calculate the area and perimeter of an isosceles triangle with a base of \(14 \text{ cm} \) and legs of \(15 \text{ cm} \) each.
2. Determine the height of an equilateral triangle with a side length of \(9 \text{ cm} \).

Answers:

1. Using the Pythagoras’ theorem, height \(h = \sqrt{15^2 – 7^2} = \sqrt{164} \approx 12.8 \text{ cm} \). Area \(= \frac{1}{2} \times 14 \times 12.8 = 89.6 \text{ cm}^2 \). Perimeter \(= 14 + 2 \times 15 = 44 \text{ cm} \).
2. Using the area formula relationship, height \(h = \sqrt{9^2 – 4.5^2} = \sqrt{56.25} = 7.5 \text{ cm} \).