8. Can the formula used to find the area of a trapezoid be used to find areas of parallelograms? What about to find areas of triangles? Use the formulas for the areas of parallelograms and triangles to justify your answer.

The formula for the area of a trapezoid is given by: \[ \text{Area} = \frac{1}{2} (b_1 + b_2) h \] where \( b_1 \) and \( b_2 \) are the lengths of the two parallel bases, and \( h \) is the height (the perpendicular distance between the bases). ### Can the trapezoid area formula be used for parallelograms? The area of a parallelogram is calculated using the formula: \[ \text{Area} = b \cdot h \] where \( b \) is the length of the base and \( h \) is the height. To see if the trapezoid formula can be adapted for parallelograms, consider a parallelogram as a special case of a trapezoid where both bases are equal (i.e., \( b_1 = b_2 = b \)). In this case, the trapezoid area formula becomes: \[ \text{Area} = \frac{1}{2} (b + b) h = \frac{1}{2} (2b) h = b \cdot h \] This shows that the trapezoid area formula can indeed be used to find the area of a parallelogram, as it simplifies to the standard formula for the area of a parallelogram. ### Can the trapezoid area formula be used for triangles? The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} b h \] where \( b \) is the base and \( h \) is the height. To relate this to the trapezoid area formula, consider a triangle as a special case of a trapezoid where one of the bases has a length of zero. If we take a triangle with base \( b \) and height \( h \), we can think of it as a trapezoid where one base is \( b \) and the other base is \( 0 \). Plugging these values into the trapezoid area formula gives: \[ \text{Area} = \frac{1}{2} (b + 0) h = \frac{1}{2} b h \] This shows that the trapezoid area formula can also be used to find the area of a triangle, as it simplifies to the standard formula for the area of a triangle. ### Conclusion In summary, the formula for the area of a trapezoid can be adapted to find the areas of both parallelograms and triangles by considering special cases where the bases are equal (for parallelograms) or one base is zero (for triangles). Thus, the trapezoid area formula is versatile and can be applied to these other shapes.