Properties, Formulas, and Example for Isosceles Triangle | Perimeter, Area, and Pythagorean Theorem

Isoceles Triangle

An isosceles triangle is a type of triangle that has two sides of equal length

An isosceles triangle is a type of triangle that has two sides of equal length. This means that two of the angles in an isosceles triangle are also equal. The side opposite the equal angles is called the base, while the other two sides are called the legs. Since the base angles are equal, they are also congruent.

Properties of an isosceles triangle:
1. Two sides are of equal length.
2. Two angles are congruent.
3. The base angles are congruent.
4. The sum of the angles in an isosceles triangle is always 180 degrees.

Formulas related to an isosceles triangle:
1. Perimeter: The perimeter is the sum of the lengths of all three sides of a triangle. For an isosceles triangle, the perimeter can be calculated as 2 times the length of one side plus the length of the base.
Perimeter = 2 × length of side + length of base

2. Area: The area of an isosceles triangle can be found by using the formula:
Area = (base × height) / 2
The height can be calculated by drawing a perpendicular line from the base to the apex of the triangle.

3. Pythagorean Theorem: If the isosceles triangle is a right triangle, meaning one of the angles is 90 degrees, then the Pythagorean theorem can be applied to find the lengths of the sides.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Example:
Let’s consider an isosceles triangle with side lengths of 5 cm, 5 cm, and the base length of 8 cm. To find the perimeter, we add up the lengths of all three sides:
Perimeter = 2 × 5 cm + 8 cm = 18 cm.

To find the area, we need to know the height. Suppose the height is 4 cm. We can apply the formula:
Area = (base × height) / 2 = (8 cm × 4 cm) / 2 = 32 cm².

Lastly, suppose the isosceles triangle is a right triangle. Using the Pythagorean theorem, we can find the length of the missing side (hypotenuse). Let’s assume the length of one leg is 7 cm and the length of the other leg is 7 cm. Then:
Hypotenuse² = (7 cm)² + (7 cm)² = 49 cm² + 49 cm² = 98 cm².
Taking the square root of both sides gives us the length of the hypotenuse, which is approximately 13.93 cm.

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