Kite and its Theorems
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In this section, we will discuss kite and its theorems.In kite, adjacent sides are equal and long diagonal bisect the small diagonal at right angle.All interior angles are acute angles.
Theorem 1 : If a quadrilateral is a kite, then its diagonals are perpendicular.
GIVEN : AB ≅ CB and AD ≅ CD
PROVE THAT : AC ⊥ BD
Proof :
| |
|
| 1)AB ≅ AD | 1) Given |
| 2) BC ≅ CD | 2) Given |
| 3) AC ≅ AC | 3) Reflexive (common side) |
| 4) ΔABC ≅ ΔADC | 4) SSS Postulates |
| 5) ∠BAE ≅ ∠DAE | 5) CPCTC |
| 6) ΔABD is an Isosceles triangle. | 6) By property of an isosceles triangle. |
| 7) ∠ABE ≅ ∠ADE | 7) Property of isosceles triangle. |
| 8) ΔABE ≅ ΔADE | 8) ASA postulate. |
| 9) ∠AEB ≅ ∠AED | 9) CPCTC |
| 10)∠AEB +∠AED = 180 | 10) Linear pair angles are supplementary. |
| 11) 2∠AEB = 180 | 11) Addition property |
| 12) ∠AEB = 90 | 12) Division property |
| 13) AC ⊥ BD and AE ⊥ BD |
13) By property of perpendicular. |
Given : ABCD is a kite with AB ≅ AD and BC ≅ CD.
Prove that : ∠A ≅ ∠C.
Proof :
| |
|
| 1)AB ≅ AD | 1) Given |
| 2) BC ≅ CD | 2) Given |
| 3) AC ≅ AC | 3) Reflexive (common side) |
| 4) ΔABC ≅ ΔADC | 4) SSS Postulates |
| 5) ∠ABC ≅ ∠ADC | 5) CPCTC |
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Example based on kite and its theorems :
In a kite, ABCD,AB = x + 2 , BC = 2x + 1. The perimeter of kite is 48cm. Find x and also find the length of each side.
Solution :
As we know that, adjacent sides in a kite are equal.
∴ AB = AD and BC = CD.
Perimeter = sum of all the sides
P = AB + BC + CD + AD
48 = x + 2 + 2x + 1 + x + 2 + 2x + 1
48 = 6x + 6
⇒ 6x = 48 -6
∴ 6x = 42
x = 42/6
x = 7
∴ AB = AD = x + 2 = 7 + 2 = 9cm
and BC = CD = 2x + 1 = 2(7) + 1 = 14 + 1 = 15 cm
Quadrilateral
• Introduction to Quadrilateral
• Types of Quadrilateral
• Properties of Quadrilateral
• Parallelogram and its Theorems
• Rectangle and its Theorems
• Square and its Theorems
• Rhombus and its Theorems
• Trapezoid (Trapezium)and its Theorems
• Kite and its Theorems
• Mid Point Theorem
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