Show that the area of a kite is half the product of its diagonals. Show this:
- (i) using algebra, and
- (ii) using geometry.
Answer: (i) Diagonal AC (= d1) splits the kite into two triangles of height d22: 2 × 12d1 × d22 = 12d1d2. (ii) The kite is half of the d1 × d2 rectangle around it.
Step-by-step solution
Idea: The diagonals of a kite cross at right angles, and the longer one (the axis of symmetry) bisects the other.
(i) using algebra, and
- Kite ABCD with AB = AD and CB = CD. A and C are each equidistant from B and D, so AC is the perpendicular bisector of BD: BD ⊥ AC at O. Let AC = d1, BD = d2, so OB = OD = d22.1 mark
- AC splits the kite into △ABC and △ADC, each with base AC = d1 and height d22. Area = 2 × 12 × d1 × d22 = 12d1d2.1 mark
12d1d2
(ii) using geometry.
- Draw the rectangle around the kite with sides parallel to the diagonals: its sides are d1 and d2, area d1d2. The diagonals cut it into 4 smaller rectangles.1 mark
- Each side of the kite is a diagonal of one small rectangle, so it cuts that rectangle into two congruent triangles: one inside the kite, one outside. So the kite is exactly half of the big rectangle: area = 12d1d2.1 mark
12d1d2
Area of a kite = ½ × d₁ × d₂ (shown by splitting along a diagonal, and by enclosing the kite in a d₁ × d₂ rectangle).
Check: Kite with diagonals 8 and 6 (cut 2 and 6 along the 8): two triangles 12 × 8 × 3 = 12 each, total 24 = 12 × 8 × 6 ✓.
Answer to write in the exam
(i)
AC ⊥ BD, OB = OD = d22
ar(△ABC) = ar(△ADC) = 12 × d1 × d22
∴ ar(kite) = 2 × 14d1d2 = 12d1d2
(ii)
Enclosing rectangle: sides d1, d2; area d1d2
Each of 4 small rectangles is halved by a side of the kite (diagonal of rectangle)
∴ ar(kite) = 12d1d2
Common mistakes that cost marks
- Assuming the diagonals bisect each other. In a kite only one diagonal is bisected.
- Using the product of the diagonals without the 12.
- Forgetting to justify that the diagonals are perpendicular.
How this can come in the exam
MCQ (1 mark)
A kite has diagonals 16 cm and 9 cm. Its area is
- 72 cm2
- 144 cm2
- 36 cm2
- 50 cm2
Show answer
(A) 72 cm2
12 × 16 × 9 = 72 cm2.
Try one yourself
A kite has area 60 cm2 and one diagonal 15 cm. Find the other diagonal.
Show answer
12 × 15 × d = 60 ⇒ d = 8 cm.
More questions like this
- Three problems about fitting congruent shapes together:
- What fraction of the triangle is shaded? What fraction of the square is shaded?
- What fraction of the rectangle is covered by the circles?
- Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!
- The figure shows nine identical rectangles fitted together to make a large rectangle whose area is 72 cm2. Find the perimeter of each small rectangle.