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A trapezoid is a unique quadrilateral, known for having two sides that are parallel. These parallel sides are called "bases". The other two sides, which may or may not be of the same length, are called "legs". The beauty of the trapezoid lies in its simplicity and the different forms it can take:

• Isosceles trapezoid: This type of trapezoid has legs that are equal in length, making symmetry another characteristic feature.
• Scalene trapezoid: Here, the legs are of different lengths.

Understanding the properties of the trapezoid, such as its angles and side lengths, is crucial when trying to solve problems related to its dimensions. In this exercise, you are particularly dealing with an isosceles trapezoid, where both legs are equal, and this helps make solving for other measurements like height much more straightforward.

The Pythagorean Theorem is a fundamental principle in geometry, particularly useful when dealing with right triangles. It connects the lengths of sides in a right triangle with the formula:
\[ a^2 + b^2 = c^2 \]Here, \(a\) and \(b\) are the lengths of the legs, and \(c\), the hypotenuse, is the side opposite the right angle. In our trapezoid problem, the theorem comes into play to find the height. By drawing a perpendicular from the top base to the bottom one, you create two right triangles.
Utilizing this tool, we can apply the theorem as follows in the exercise:

• Identify the triangle's legs, half of the longer base is one leg \(a = 5\) units in this case.
• The hypotenuse is the trapezoid's leg (13 units).

Use the theorem to solve for the third side \(h\) — the height:
\[ h = \sqrt{13^2 - 5^2} \]This step is vital as it allows us to determine the height, a crucial element for calculating the area.

Computing the area of a trapezoid involves combining the lengths of its two bases and multiplying by the height. The formula for the area of a trapezoid is:
\[ \text{Area} = \frac{1}{2} \times (B + b) \times h \]Where:

• \(B\) represents the longer base, in this case, 20 units.
• \(b\) is the shorter base, 10 units here.
• \(h\) is the height you calculate using the Pythagorean theorem.

Putting it all together, you substitute the known values into the formula to find the area:
\[ \text{Area} = \frac{1}{2} \times (20 + 10) \times \sqrt{13^2 - 5^2}\]This formula allows you to find the trapezoid's area efficiently by considering its height and base lengths. This approach underscores why understanding all trapezoid components and how to use them forms the foundation for such geometric calculations.