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The Pythagorean theorem is a powerful tool in geometry that allows us to relate the sides of a right triangle. In this problem involving an isosceles trapezium, we make use of the Pythagorean theorem to calculate the length of the longer parallel side. Here’s how it works:
- Identify a right triangle within the trapezium, formed when you drop a perpendicular from the shorter parallel side to the non-parallel side.- The perpendicular (height of the trapezium) serves as one leg of the triangle, the hypotenuse is the slanted side (20 cm), and the other leg is part of the base of the trapezium.
Using the formula, \[ a^2 + b^2 = c^2 \]where:

• \( a \) is the leg \( h \) (height)
• \( b \) is the horizontal leg \( \frac{b-30}{2} \)
• \( c \) is the hypotenuse \( 20 \)

we derive the equation:\[ \left(\frac{b-30}{2}\right)^2 + h^2 = 20^2 \]This equation is the key to finding the unknown side of the trapezium.

Calculating the area of an isosceles trapezium requires understanding the trapezium area formula. The area formula for a trapezium considers both the parallel sides and the height, given by:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]where:

• \( b_1 \) is the shorter parallel side (30 cm)
• \( b_2 \) is the longer parallel side
• \( h \) is the height

Substituting in the known values allows us to use the calculated length from the Pythagorean theorem solution for the longer parallel side \( b_2 \).
Thus, we get:\[ A = \frac{1}{2} \times (30 + b) \times h \]Simplifying it further using \( b = 30 + 2\sqrt{400-h^2} \), we derive the trapezium’s area as:\[ h(30 + \sqrt{400-h^2}) \].This concise formula provides a direct way to compute the area given any height \( h \).

Solving a geometry problem involves several steps that require logic and a clear understanding of shapes and their properties. In the context of the isosceles trapezium, here is a simple approach to problem-solving:
- **Visualize the Problem**: Sketch the trapezium and label all known quantities. - **Identify Relevant Theorems**: For this problem, focus on the Pythagorean theorem. - **Set up Equations**: Use given dimensions to formulate mathematical equations.
After deriving equations based on the problem's geometry, solve them to find unknowns, such as the length of a side or the area. Geometry problem-solving often requires:

• Breaking down complex shapes into simpler components (such as right triangles)
• Systematically applying formulas (e.g., area or side-length using the Pythagorean theorem)
• Checking that solutions make sense with respect to the initial question

With practice, problem-solving can become a systematic approach governed by step-by-step logic.

An isosceles trapezium inherits properties from isosceles triangles, particularly due to its pair of equal-length non-parallel sides. Recognizing these properties can simplify your approach:
- **Symmetry**: This means both angles at the base of each leg are equal. - **Equal Non-Parallel Sides**: In our example, these sides measure 20 cm each. - **Use in constructing right triangles**: Constructing a right triangle using these sides assists in applying the Pythagorean theorem.
The knowledge of isosceles triangle properties helps in deconstructing the trapezium:

• It ensures balance and proportionality in the figure, easing calculations.
• It allows for creating right triangles that simplify determining unknown dimensions.
• Understanding these properties assists in visualizing symmetry and solving area-related questions more easily.

Recognizing these inherent geometric properties reduces the complexity of problems involving isosceles trapeziums.