Which expression gives the area of the trapezoid as the sum of the three triangles' areas? \begin{tabular}{|l}\( 3\left(\frac{1}{2} a b\right) \) \\ \( \frac{1}{2}\left(a b+c^{2}\right) \) \\ \( \frac{1}{2} a(a+b) \) \\ Submit \( c^{2} \)\end{tabular} \[ \begin{array}{l}\text { Workitout }\end{array} \]

To find the expression that gives the area of a trapezoid as the sum of the areas of three triangles, we need to understand the geometry of a trapezoid and how its area can be represented. ### Step 1: Understanding the trapezoid A trapezoid has two parallel sides, which we can denote as \( a \) (the length of the top base) and \( b \) (the length of the bottom base). The height of the trapezoid is denoted as \( h \). ### Step 2: Area of the trapezoid The area \( A \) of a trapezoid can be calculated using the formula: \[ A = \frac{1}{2} (a + b) h \] ### Step 3: Dividing the trapezoid into triangles To express the area of the trapezoid as the sum of the areas of three triangles, we can visualize the trapezoid being divided into three triangles. 1. **Triangle 1**: Formed by the height from the top base \( a \) to the height \( h \). 2. **Triangle 2**: Formed by the height from the bottom base \( b \) to the height \( h \). 3. **Triangle 3**: A triangle that connects the two bases. ### Step 4: Area of the triangles The area of each triangle can be expressed as follows: - Area of Triangle 1: \( \frac{1}{2} a h \) - Area of Triangle 2: \( \frac{1}{2} b h \) - Area of Triangle 3: The area can be represented in terms of the height and the difference in the bases. ### Step 5: Summing the areas The total area of the trapezoid can be represented as: \[ A = \frac{1}{2} a h + \frac{1}{2} b h + \text{Area of Triangle 3} \] ### Step 6: Evaluating the options Now, let's evaluate the given options to see which one can represent the area of the trapezoid as the sum of the areas of three triangles. 1. **Option 1**: \( 3\left(\frac{1}{2} a b\right) \) - This does not represent the area of the trapezoid. 2. **Option 2**: \( \frac{1}{2}(a b + c^{2}) \) - This does not seem to relate to the area of the trapezoid. 3. **Option 3**: \( \frac{1}{2} a(a+b) \) - This does not represent the area of the trapezoid. 4. **Option 4**: \( c^{2} \) - This does not represent the area of the trapezoid. ### Conclusion None of the provided options directly represent the area of the trapezoid as the sum of the areas of three triangles based on the standard geometric interpretation. If we were to derive a specific expression for the area of the trapezoid based on the triangles, we would need to know more about the dimensions and relationships between the bases and height. Thus, the answer is that none of the expressions provided correctly give the area of the trapezoid as the sum of the areas of three triangles.