Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. Students learn through exploration that the area of a triangle is exactly half of the area of its corresponding rectangle. They extend their previous knowledge about the area formula for rectangles to evaluate the area of the rectangle using A = bh and discover through manipulation that the area of a right triangle is exactly half that of its corresponding rectangle.
Students discover that triangles have altitude, which is the length of the height of the triangle. The altitude is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of the triangle can be a base, but the altitude always determines the base. They move from recognizing right triangles as categories to determining that right triangles are constructed when altitudes are perpendicular and meet the base at one side.
Acute triangles are constructed when the altitude is perpendicular and meets within the length of the base, and obtuse triangles are constructed when the altitude is perpendicular and lies outside the length of the base. Students use this information to cut triangular pieces and rearrange them to fit exactly within one half of the corresponding rectangle to determine that the area formula for any triangle can be determined using A = 1/2(bh).
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