Two line segments can be drawn to intersect in various ways — they could be perpendicular, they could bisect one another, or both. Depending on the arrangement of these segments, various quadrilaterals can be formed by connecting the endpoints of the segments.

This activity allows you to discover what types of quadrilaterals can be formed when the diagonals meet in various ways.

### Instructions

• Use the selectors to choose Perpendicular Diagonals, Congruent Diagonals, and the number of Bisected Diagonals.
Note that it is possible to select multiple options, but some options are incompatible — for instance, Congruent Diagonals and 1 Bisected cannot be chosen simultaneously.
• A quadrilateral will be created with the selected characteristics. All vertices and points of intersection will be colored while, orange, or black:
• white – the point can be moved freely
• orange – the point can only be moved along the line segment on which it lies
• black – the point cannot be moved
• The question mark button (?) can be used to reveal the various types of shapes that are possible for the given characteristics.
• The Camera button can be used to capture up to four images, which will be displayed in the gallery below the work area; the corresponding trash can icon can be used to delete each image.
• You can Reset the shape at any point.

### Exploration

Select Perpendicular Diagonals, but make sure that Congruent Diagonals and Bisected Diagonals are not selected. Drag the orange and white points to change the size and shape of the quadrilateral (but notice that the diagonals always meet at a right angle). What types of quadrilaterals can be formed such that the diagonals are perpendicular?
• Can a rectangle be formed?
• Can a trapezoid be formed?
• Can any other shapes be formed?

Use the ? button to reveal the types of quadrilaterals that can be formed with perpendicular diagonals. What shapes can be formed if other characteristics are selected?

### Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• High School (9-12)
• Geometry