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K I t e
Definition• KITE – is a quadrilateral with two sets of distinct
adjacent congruent sides, but opposite sides are not congruent.
CHINA
IN ANCIENT TIME
KITE were widely considered to be useful for ensuring a good harvest or scaring away evil spirits.
IN MODERN TIME
KITE became more widely known as children's toys and came to be used primarily as a leisure activity
• From the definition, a kite is the only quadrilateral that we have discussed that could be concave or non convex. Concave or non convex kite is a kite whose diagonal do not intersect. If a kite is concave or non convex, it is called a dart .
•CONVEX KITE-
D
C
A
B
the diagonals of a kite intersect.
• The angles between the congruent sides are called vertex angles . The other angles are called non-vertex angles . If we draw the diagonal through the vertex angles, we would have two congruent triangles.
B
A C D
THEOREM 1: The non-vertex angles of a kite are congruent and the diagonal through the vertex angle is the angle bisector for both angles.
PROOF:
GIVEN: KITE WITH ≅ AND ≅
STATEMENTS REASONS
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
KITE WITH ≅ AND ≅ GIVEN
REFLEXIVE PROPERTY
SSS CONGRUENCE POSTULATE
CPCTC
AND CPCTC
PROVE: , AND
THEOREM 2: The diagonals of a kite are perpendicular to each other.
PROOF:
GIVEN: Kite BCDA
STATEMENTS REASONS
1. 1.
2. 2.
3. 3.
4. 4.
D
C
A
B
Kite BCDA GIVEN≅ AND ≅ Definition of kiteDefinition of congruent segments AND
𝐶𝐴⊥𝐵𝐷 If a line contains two points each of which is equidistant from the endpoints of a segment, then the line is perpendicular bisector of the segment.
PROVE:
Theorem 3: The area of a kite is half the product of the lengths of the diagonals.
w
PROOF:
GIVEN: Kite BCDA
STATEMENTS REASONS
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
Kite BCDA GIVEN
𝐶𝐴⊥𝐵𝐷 The diagonals of a kite are perpendicular to each other.
𝑨𝒓𝒆𝒂𝒐𝒇 𝒌𝒊𝒕𝒆𝑩𝑪𝑫𝑨=𝑨𝒓𝒆𝒂𝒐𝒇 ∆𝑩𝑪𝑨+𝑨𝒓𝒆𝒂𝒐𝒇 ∆𝑪𝑫𝑨 Area addition postulate
Area formula for triangles
𝐀 𝒓𝒆𝒂𝒐𝒇 𝒌𝒊𝒕𝒆𝑩𝑪𝑫𝑨=𝟏𝟐
(𝑪𝑨 ) (𝑩𝑾 )+𝟏𝟐
(𝑪𝑨)(𝑫𝑾 )Substitution
𝐀 𝒓𝒆𝒂𝒐𝒇 𝒌𝒊𝒕𝒆𝑩𝑪𝑫𝑨=𝟏𝟐
(𝑪𝑨 ) (𝑩𝑾 +𝑫𝑾 ) Associative Property
𝐵𝑊 +𝐷𝑊=𝐵𝐷 Segment Addition Postulate
𝐀 𝒓𝒆𝒂𝒐𝒇 𝒌𝒊𝒕𝒆𝑩𝑪𝑫𝑨=𝟏𝟐
(𝑪𝑨 )(𝑩𝑫) Substitution
PROVE:
Example 1
•Find the area of the kite WXYZ.
20
12
1212
UW
Z
Y
X
Example1 Continued
20
12
1212
UW
Z
Y
X
We can now use the formula in finding the area of the kite.
Area of kite WXYZ=
Area of kite WXYZ=
Area of kite WXYZ=
Area of kite WXYZ=
EXAMPLE 2: Given kite WXYZ
20
12
1212
UW
Z
Y
X
9
9
What is the length of segment XY?
EXAMPLE 2: Given kite WXYZ20
12
1212
UW
Z
Y
X
9
9
𝑋𝑌 2=𝑈𝑋 2+𝑈𝑌 2
𝑋𝑌 2=92+122
𝑋𝑌 2=81+144
𝑋𝑌 2=225
XY
Example 3
• Find mG and mJ.
60132
J
G
H K
Since GHJK is a kite G J
So 2(mG) + 132 + 60 = 360
2(mG) =168
mG = 84 and mJ = 84
Try This!• RSTU is a kite. Find mR, mS and mT.
x
125
x+30
S
U
R T
x +30 + 125 + 125 + x = 360
2x + 280 = 360
2x = 80
x = 40
So mR = 70, mT = 40 and mS = 125
QUIZ• Given kite BCDA and point P be the point of
intersection of the diagonals , consider the given information below and answer the question that follows.
1. 2. What is the area of kite BCDA? 3. What is the area of kite BCDA?
D
C
A
B
