91Ó°ÊÓ

An isosceles triangle is defined as a type of triangle with at least two sides of equal length. These equal sides are called the legs, and the other is referred to as the base. In our exercise, the sides that measure 10 are the legs, and the side measuring 16 is the base.

• One of the unique properties of isosceles triangles is their symmetry. The two equal sides mean that the triangle is symmetrical around the altitude (height) from the base to the opposite vertex.
• The angles opposite the equal sides are also equal, which can be helpful in various calculations.

Understanding these properties helps in identifying and solving problems related to isosceles triangles.
They are crucial when determining angles or other lengths within the triangle.

The Pythagorean theorem is a fundamental principle used to relate the sides of a right triangle. It states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). Mathematically, it is expressed as:

• \(a^2 + b^2 = c^2\)

In our case, when the height is drawn in the isosceles triangle, it creates two right-angled triangles. Here, the hypotenuse is 10 (the length of one leg of the isosceles triangle), and one of the legs is 8 (half of the base).
To find the missing height (other leg), we use the formula:

• \(h = \sqrt{c^2 - b^2} = \sqrt{10^2 - 8^2} = 6\)

The Pythagorean theorem is an essential tool that allows us to find unknown side lengths in right-angled triangles efficiently.

Calculating the area of a triangle is a standard mathematical procedure, particularly useful in geometry. The area of any triangle can be determined using the formula:

• \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)

For the isosceles triangle in the exercise, once we find the height using the Pythagorean theorem, we can substitute the values into the area formula.

• The base is 16, and the height is 6.
• Thus, the area is \(\frac{1}{2} \times 16 \times 6 = 48\)

This method applies not only to isosceles triangles but any triangle where the base and height can be determined. Being thorough with this formula is crucial as it frequently comes up in geometry and various applications.