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Explore the Midpoint Quadrilateral of a Quadrilateral Explore the Midpoint Quadrilateral of a Quadrilateral

Prerequisites and Objectives

>> Students know special quadrilaterals such as rectangle, square, trapezoid, kite, rhombus, and parallelogram.

>> Students construct the midpoints of the sides of a quadrilateral and investigate the midpoint quadrilateral.

>> Students explain why this quadrilateral is a parallelogram. sketchometry Instructions

Students should know

>> how to construct a quadrilateral/polygon,

>> how to construct the midpoint of a line segment,

>> how to show grid lines and

PropertiesShow Grid

>> how to snap to the grid.

PropertiesSnap to grid

Further Explorations

>> Students compare this result with the midpoint quadrilateral of a kite.

>> Students investigate midpoint quadrilaterals of other special quadri- laterals. Explore the Midpoint Quadrilateral of a Quadrilateral

Construction

>> Choose any four points and connect them to obtain a quadrilateral.

>> Construct the midpoints E, F, G, and H of the sides of the quadrilateral.

>> Draw the midpoint quadrilateral EFGH.

Exploration

>> Drag any of the vertices A, B, C, and D and observe the midpoint quadri- lateral EFGH. Describe its shape.

>> Write down your observation in your study journal.

>> Support your conjecture with a proof. Hint: Draw the diagonals of quadri- lateral ABCD as auxiliary lines.

>> Draw special types of quadrilaterals ABCD (square, rectangle, parallelo- gram, rhombus) by placing the vertices of these quadrilaterals on grid points. Describe what kind of midpoint quadrilaterals you get.

>> Write your results in your study journal together with sketches and proofs.

sketchometry.org – Center for Mobile Learning with Digital Technology – University of Bayreuth – Result Sheet

Explore the Midpoint Quadrilateral of a Quadrilateral

>> Which special shape does the midpoint quadrilateral of a quadrilateral have? Write down your conjecture.

>> Draw the diagonals of the quadrilateral ABCD. Consider the triangles ACD and CAB. Then consider the triangles DBC and BDA. Write down a proof of your above conjecture.

>> Draw special types of quadrilaterals (square, rhombus, parallelogram, rectangle) by placing the vertices of the quadrilateral ABCD on grid points. Describe what kind of midpoint quadrilaterals you get.

sketchometry.org – Center for Mobile Learning with Digital Technology – University of Bayreuth