Properties of Kites
A kite has two pairs of consecutive congruent sides. Its diagonals are perpendicular, and exactly one diagonal is bisected by the other.
Introduction
Kite: AB = AD, CB = CD; diagonals perpendicular
Past Knowledge
Rhombus properties (9.2.4). Perpendicular lines. Triangle congruence (5.2).
Today's Goal
Prove and apply kite properties: ⊥ diagonals, one pair of ≅ angles.
Future Success
Coordinate proofs (9.3.4), area formulas, quadrilateral classification.
Key Concepts
Kite Properties
- Two pairs of consecutive congruent sides (not opposite)
- Diagonals are perpendicular
- The main diagonal (connecting the two vertex angles) bisects the other diagonal
- The main diagonal bisects the vertex angles
- Exactly one pair of opposite angles is congruent (the non-vertex angles)
Area of a Kite
Same formula as a rhombus — because both have ⊥ diagonals
Kite vs. Rhombus
- Kite: 2 pairs of consecutive ≅ sides, diagonals ⊥, only one diagonal bisected
- Rhombus: 4 ≅ sides, a parallelogram, BOTH diagonals bisected, diagonals ⊥
- A rhombus is a special kite where both pairs are equal to each other
Theorem & Proof
Two-Column Proof: Diagonals of a Kite are Perpendicular
Given: Kite with and
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | and | Given (definition of kite) |
| 2 | and are each equidistant from and | From step 1 |
| 3 | is the perpendicular bisector of | If a point is equidistant from two endpoints, it lies on the ⊥ bisector of that segment. Two such points determine the ⊥ bisector. |
∎ Points A and C are both equidistant from B and D, so AC must be the perpendicular bisector of BD.
Worked Examples
Area of a Kite
Kite with diagonals 12 and 18. Find the area.
Area = 108 sq units
Finding Angles
Kite ABCD has ∠A = 110° (vertex angle), ∠C = 50° (vertex angle). Find ∠B and ∠D.
Angle sum = 360°:
∠B = ∠D (non-vertex angles are congruent): →
∠B = ∠D = 100°
Side Lengths from Diagonals
Kite ABCD: main diagonal AC = 16 splits BD into segments of 5 each (BE = ED = 5). The shorter portion of AC from the intersection to A is 6. Find all side lengths.
Let E be the intersection. AE = 6, EC = 10, BE = ED = 5.
Common Pitfalls
Thinking Both Diagonals Are Bisected
Only the “cross” diagonal (connecting the non-vertex angles) is bisected. The main diagonal (through the vertex angles) is NOT bisected. If both diagonals were bisected, it would be a parallelogram.
Confusing Consecutive vs. Opposite Sides
A kite has consecutive sides congruent. A parallelogram has opposite sides congruent. A kite is NOT a parallelogram.
Real-Life Applications
Actual Kites!
Flying kites are shaped as geometric kites because the perpendicular diagonals create four right triangles, maximizing surface area while minimizing the framing needed.
Aircraft Wing Design
Some delta-wing aircraft use kite-shaped wing cross-sections. The perpendicular diagonal structure provides excellent strength-to-weight ratio.
Practice Quiz
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