Special Types of Quadrilaterals
172. The Kite's Balance · adjacent sides equal, diagonals perpendicular
Diagonals are always perpendicular, and AC always bisects BD.
A kite is a quadrilateral with two pairs of adjacent sides equal. Its diagonals are perpendicular to each other, and the longer diagonal bisects the shorter at right angles.
What this lesson covers
Try to break it
Drag A up and C down to set the vertical diagonal; drag B and D out to set the horizontal half-diagonals. As long as B and D stay equidistant from the centre, AB = AD and CB = CD and the kite keeps its symmetry. Pull B and D to unequal distances and the kite breaks — the adjacent-sides equality fails.
How you build it
Construct a kite from perpendicular diagonals.
- Draw segment BD, the horizontal diagonal of the kite.
- Construct the perpendicular bisector of BD; it crosses BD at O.
- Mark point A on the bisector above BD.
- Mark point C on the bisector below BD, with OC different from OA.
- Draw segment AB of the kite.
- Draw segment BC of the kite.
- Draw segment CD of the kite.
- Draw segment DA to complete the kite.
The proof, step by step
Prove that the diagonals of a kite are perpendicular and one bisects the other.
- In ΔABD, AB = AD (given). Thus, ΔABD is isosceles.
- In ΔCBD, CB = CD (given). Thus, ΔCBD is isosceles.
- Points A and C are both equidistant from B and D, so AC is the perpendicular bisector of BD.
- Therefore, AC ⊥ BD and OB = OD.
- By SSS, ΔABC ≅ ΔADC, giving ∠ABC = ∠ADC and AC bisects ∠BAD and ∠BCD.
Worked example
In a kite ABCD, diagonals intersect at O. If AC = 10 cm and BD = 8 cm, what is the area of the kite?
Area of a kite = ½ × d₁ × d₂ = ½ × 10 × 8 = 40 cm².
- 20 cm²
- 40 cm² — correct
- 80 cm²
- 160 cm²