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