Kites in Geometry
Kite definition geometry
You probably know a kite as that wonderful toy that flies aloft on the wind, tethered to you by string. That toy kite is based on the geometric shape, the kite.
A kite is a quadrilateral shape with two pairs of adjacent (touching), congruent (equal-length) sides. That means a kite is all of this:
A plane figure
A closed shape
A polygon
Sometimes a kite can be a rhombus (four congruent sides), a dart, or even a square (four congruent sides and four congruent interior angles).
Some kites are rhombi, darts, and squares. Not every rhombus or square is a kite. All darts are kites.
Kites can be convex or concave. A dart is a concave kite. That means two of its sides move inward, toward the inside of the shape, and one of the four interior angles is greater than 180°. A dart is also called a chevron or arrowhead.
How to construct a kite in geometry
You can make a kite. Find four uncooked spaghetti strands. Cut or break two spaghetti strands to be equal to each other, but shorter than the other two strands.
Touch two endpoints of the short strands together. Touch two endpoints of the longer strands together. Now carefully bring the remaining four endpoints together so an endpoint of each short piece touches an endpoint of each long piece. You have a kite!
How to draw a kite in geometry
You can also draw a kite. Use a protractor, ruler and pencil. Draw a line segment (call it KI) and, from endpoint II, draw another line segment the same length as KI. That new segment will be IT.
The angle those two line segments make (∠I) can be any angle except 180° (a straight angle).
Draw a dashed line to connect endpoints K and T. This is the diagonal that, eventually, will probably be inside the kite. Now use your protractor. Line it up along diagonal KT so the 90° mark is at ∠I. Mark the spot on diagonal KT where the perpendicular touches; that will be the middle of KT.
Lightly draw that perpendicular as a dashed line passing through ∠I and the center of diagonal KT. Make that line as long as you like.
If you end the line closer to ∠I than diagonal KT, you will get a dart. If you end the new line further away from ∠I than diagonal KT, you will make a convex kite.
Connect the endpoint of the perpendicular line with endpoint T. Label it point E. Connect point E with point K, creating line segment EK. Notice that line segments (or sides) TE and EK are equal. Notice that sides KI and IT are equal.
You probably drew your kite so sides KI and EK are not equal. That also means IT and TE are not equal. You could have drawn them all equal, making a rhombus (or a square, if the interior angles are right angles).
Properties of kites
The kite's sides, angles, and diagonals all have identifying properties.
Kite sides
To be a kite, a quadrilateral must have two pairs of sides that are equal to one another and touching. This makes two pairs of adjacent, congruent sides.
You could have one pair of congruent, adjacent sides but not have a kite. The other two sides could be of unequal lengths. Then you would have only a quadrilateral.
Your kite could have four congruent sides. Your quadrilateral would be a kite (two pairs of adjacent, congruent sides) and a rhombus (four congruent sides).
Some (but not all) kites are rhombi. If your kite/rhombus has four equal interior angles, you also have a square.
Kite angles
Where two unequal-length sides meet in a kite, the interior angle they create will always be equal to its opposite angle. Look at the kite you drew.
∠K=∠T and ∠I =∠E
It is possible to have all four interior angles equal, making a kite that is also a square.
Kite diagonals
The two diagonals of our kite, KT and IE, intersect at a right angle. In every kite, the diagonals intersect at 90°. Sometimes one of those diagonals could be outside the shape; then you have a dart. That does not matter; the intersection of diagonals of a kite is always a right angle.
A second identifying property of the diagonals of kites is that one of the diagonals bisects, or halves, the other diagonal. They could both bisect each other, making a square, or only the longer one could bisect the shorter one.
Lesson summary
For what seems to be a really simple shape, a kite has a lot of interesting features. Using the video and this written lesson, we have learned that a kite is a quadrilateral with two pairs of adjacent, congruent sides.
Kites can be rhombi, darts, or squares. We also know that the angles created by unequal-length sides are always congruent.
Finally, we know that the kite's diagonals always cross at a right angle and one diagonal always bisects the other.