now the net shape that we need to talk
about is the kite so let's call this a B
C and D so the first thing needs to know
is that a B and a D
are congruent next you need to know that
BC and DC are congruent as well and then
also one pair of opposite angles are
congruent so angle B and angle D are
congruent to each other now let's focus
on the diagonals the two diagonals they
meet at right angles so they're
perpendicular to each other and also AC
it bisects angle a into two congruent
angles so these angles are congruent to
each other
it also bisects angle C into two
congruent angles now angle a and angle C
are not congruent to each other in this
example in addition AC let's fix that AC
is the perpendicular bisector of BD so
that means that let's call this point D
first so E is the midpoint of BD which
means B E is congruent to Edie so those
are some basic properties of kites
and we'll call this D one the left of
diagonal one and this is V 2 like a
rhombus the area of a kite is one-half
of D 1 times D 2 so this is just a basic
introduction to kites and so those are
some properties that you want to be
familiar with now let's work on some
problems with kites
so let's say that angle a is 60 degrees
and angle C is a hundred what is the
measure of angle B so feel free to try
this problem now if we call angle B is
congruent to angle D the part of this
short diagonal and so they're going to
be equal to each other so therefore if
we call angle B X angle D has the X and
the kite is basically a quadrilateral a
four-sided polygon and the total
interior angle of a four-sided polygon
is 360 so we can say that 60 plus 100
plus X and X which we can represent as
2x they have to add up to 360 so 60 plus
100 is 160 and if we subtract both sides
by 160 360 minus 160 is 200 so therefore
X is 200 divided by 2 so X is a hundred
degrees which means angle B is a hundred
degrees now let's look at a second
example
so let's call this a B C and D and let's
draw the diagonals for this problem
we're gonna call this e so let's say
that a E is equal to 6 and B E is 8 and
E C is 15 calculate the area of this
kite and also the perimeter of the kite
so let's focus on the area D 1 is
basically a sea it's 6 plus 15 so D 1 is
21 units long now notice that E is the
midpoint of BD so AC by sex and BD into
two congruent parts which means that B E
and E G are the same so if B E is 8 eg
is 8 which means that the second
diagonal is 8 plus 8 or 16 units long so
now we can calculate the area of the
kite so the area is 1/2 of D 1 times D 2
so D 1 in this example is 21 D 2 is 16
now half of 16 is 8 so then we have 8
times 21 8 times 20 if you have 8 $20
bills that's 160 bucks and 8 times 1 is
8 so 8 times 21 is 168 so that's the
area of this particular type now let's
focus on calculating the perimeter of
the kite
so keep in mind the two diagonals they
meet at right angles so this is the
90-degree angle which means there's four
right triangles within this kite so
let's focus on this triangle triangle B
EC so this side is 8 this is 15 what's
the hypotenuse now there are some
special triangles that you want to keep
in mind there's the 3 4 5 right triangle
the 5 12 13 there's a 7 24 25 triangle
and 8 15 17 triangle so the hypotenuse
is 17 and you can calculate it let's say
if we call this H if you did H squared
is equal to a squared plus B squared 8
squared is 64 15 squared is 225 and then
64 plus 225 that's 289 and the square
root of 289 is 17 so BC is 17 now we
know that BC and DC are congruent so DC
is also 17 so now we got to find a B
which is congruent to ad so notice that
if we take these numbers and multiply by
2 this will give us the 6 8 10 triangle
so this is 6 and that's 8 the hypotenuse
must be 10 which means this is 10 so the
perimeter is the sum of all four sides
so it's 10 plus 10 plus 17 plus 17 10
plus 10 is 20 17 plus 17 is 34 so the
perimeter of this figure is 54 units now
let's work on another problem so let's
say if we have a kite that looks like
this and let's use the same letters to
describe it and this is going to be the
short diagonal and here we have the long
diagonal now let's say that let's call
this a first
so let's say a B II this angle is 35
degrees and angle CDE
is 25 calculate every other angle in his
figure now we know that the two
diagonals meet at right angles so this
is 90 and everything else is Nadia
around it and this is 90 as well now if
this is 35 and this is 90 what's angle
BAE the three interior angles of a
triangle must add up to 180
so BAE must be 180 minus 90 minus 35 180
minus 90 is 90 and 90 minus 35 90 minus
30 is 60 and 60 minus 5 is 55 so this
angle is 55 now these two angles are
congruent so therefore this must be 25
and these two are congruent as well so
this is 35 now these angles as a whole
are congruent but they're not bisected
into two congruent angles the long
diagonal bisects these angles into two
congruent angles now if this is 25 and
this is 90 what's the missing angle here
so we know it has to be 180 minus 90
minus 25 so this is 90 minus 25 90 minus
20 is 70 and 70 minus 5 is 65 so this is
65 and this is 65 which means this has
to be 55 because these three have to add
up to 180 so as you can see these two
angles they're congruent and these two
angles are congruent as well and these
two angles as a whole are through and
they're not bisected into two equal
angles but they both add up to 120 so
they're congruent as a whole and so
that's it for the angles within a kite
let's try one more problem
so this is a B C D and E so in this
problem B E is equal to 4x plus 1 and E
D is equal to 6x minus 9 and let's say
that a E is equal to x squared plus 10x
minus 3 so go ahead and determine the
measure of a B so that's the goal in the
problem now E is the midpoint of BD so
that means that B e and E D is congruent
so we could set B e and E D equal to
each other
so therefore 4x plus 1 is equal to 6x
minus 9 so let's subtract both sides by
4x and let's add 9 to both sides so
these will cancel 1 plus 9 is 10 6 X
minus 4 X is 2x and so if we divide both
sides by 2 we can see that X is 10
divided by 2 which is 5 so now that we
have the value of x we can calculate the
length of segment AE so AE is x squared
plus 10x minus 3 and if we replace X
with 5 we're gonna have 5 squared plus
10 times 5 minus 3 so 5 times 5 is 25 10
times 5 is 15 and 25 plus 50 is 75 and
75 minus 3 is 72
so AE is 72 in order to find a B we need
to find the value of B and so we know
that B E is 4x plus 1 so that's gonna be
4 times 5 plus 1 so 4 times 5 is 20 plus
1 that's 21
so now let's calculate the hypotenuse of
that triangle since we know this is a
right angle so let's call this C C
squared is equal to a squared plus B
squared and so a is 72 B is 21 72
squared is 5 thousand 184 21 squared is
441 and so that's equal to C squared so
51 84 plus 441 that's 56 25 now let's
take the square root of both sides so
the hypotenuse which is the measure of a
B that's equal to 75 and so that's the
answer
