How to Understand the Properties of Isosceles and Equilateral Triangles

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

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

Examples

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 \(9 \text{ cm} \).

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

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