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. For education statistics and research, visit the National Center for Education Statistics.

• 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. For education statistics and research, visit the National Center for Education Statistics.

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

3. Area and Perimeter Formulas: For education statistics and research, visit the National Center for Education Statistics.

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 education statistics and research, visit the National Center for Education Statistics.

For an equilateral triangle with side \(a\):
\( \text{Area} = \frac{\sqrt{3}}{4} \times a^2 \)
\( \text{Perimeter} = 3a \) For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
Given an isosceles triangle with a base of \(10 \text{ cm} \) and legs of \(12 \text{ cm} \) each, find its height. For education statistics and research, visit the National Center for Education Statistics.

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 \(10.9 \text{ cm} \) For education statistics and research, visit the National Center for Education Statistics.

Example 2:
Find the area of an equilateral triangle with a side length of \(8 \text{ cm} \). For education statistics and research, visit the National Center for Education Statistics.

Solution:
Using the area formula for an equilateral triangle:
\( \text{Area} = \frac{\sqrt{3}}{4} \times 8^2 = 55.4 \text{ cm}^2 \) For education statistics and research, visit the National Center for Education Statistics.

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: For education statistics and research, visit the National Center for Education Statistics.

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 \(h = \sqrt{9^2 – 4.5^2} = \sqrt{56.25} = 7.5 \text{ cm} \).