To find the area of a trapezoid by decomposing it into rectangles and triangles, let's consider a general trapezoid with bases \( b_1 \) and \( b_2 \) (where \( b_1 \) is the longer base) and height \( h \).
Here’s a step-by-step method to calculate the area:
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Identify the shapes: A trapezoid can be decomposed into a rectangle and two triangles. The rectangle can be formed by drawing a line parallel to the bases from the shorter base to the longer base at the height \( h \) of the trapezoid.
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Calculate the area of the rectangle: If we assume the trapezoid's height is \( h \), and the shorter base \( b_2 \) is equal to one of the rectangle's sides, the area \( A_{rectangle} \) of the rectangle is given by: \[ A_{rectangle} = b_2 \times h \]
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Calculate the areas of the triangles: The areas of the two triangles can be computed using: \[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
The base of each triangle can be determined by:
- The left triangle’s base is \((b_1 - b_2)/2\)
- The height remains \( h \)
Therefore, the area of one triangle (assuming both triangles are the same) is: \[ A_{triangle} = \frac{1}{2} \times \frac{(b_1 - b_2)}{2} \times h \]
Since there are two identical triangles, we double this area: \[ \text{Area of both triangles} = 2 \times \left( \frac{1}{2} \times \frac{(b_1 - b_2)}{2} \times h \right) = \frac{(b_1 - b_2) \times h}{2} \]
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Combine areas: The total area of the trapezoid \( A_{trapezoid} \) is given by adding the area of the rectangle and the two triangles: \[ A_{trapezoid} = A_{rectangle} + A_{triangle} = (b_2 \times h) + \left( \frac{(b_1 - b_2) \times h}{2} \right) \] \[ A_{trapezoid} = b_2 \times h + \frac{(b_1 - b_2) \times h}{2} = h \left(b_2 + \frac{(b_1 - b_2)}{2}\right) = h \left(\frac{2b_2 + b_1 - b_2}{2}\right) = h \left(\frac{b_1 + b_2}{2}\right) \]
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Final formula for trapezoid area: The final area formula becomes: \[ A_{trapezoid} = \frac{(b_1 + b_2) \times h}{2} \]
Using this method, you can find the area of the trapezoid by substituting the values for \( b_1 \), \( b_2 \), and \( h \). If you have specific dimensions for the trapezoid you want to calculate, please provide them, and I can assist you with the exact area calculation.