So quadrilateral ABCD, they’re

telling us it is a rhombus, and what we need to

do, we need to prove that the area of this rhombus is

equal to 1/2 times AC times BD. So we’re essentially proving

that the area of a rhombus is 1/2 times the product of

the lengths of its diagonals. So let’s see what

we can do over here. So there’s a bunch of

things we know about rhombi and all rhombi are

parallelograms, so there’s tons of things that

we know about parallelograms. First of all, if it’s a rhombus,

we know that all of the sides are congruent. So that side length is

equal to that side length is equal to that side length is

equal to that side length. Because it’s a

parallelogram, we know the diagonals bisect each other. So we know that

this length– let me call this point

over here B, let’s call this E. We know that BE

is going to be equal to ED. So that’s BE, we know that’s

going to be equal to ED. And we know that

AE is equal to EC. We also know, because

this is a rhombus, and we proved this

in the last video, that the diagonals, not only

do they bisect each other, but they are also perpendicular. So we know that this

is a right angle, this is a right angle,

that is a right angle, and then this is a right angle. So the easiest way

to think about it is if we can show that

this triangle ADC is congruent to triangle ABC,

and if we can figure out the area of one of them,

we can just double it. So the first part is

pretty straightforward. So we can see that

triangle ADC is going to be congruent

to triangle ABC, and we know that by

side-side-side congruency. This side is congruent

to that side. This side is congruent to

that side, and they both share a C right over here. So this is by side-side-side. And so we can say that the

area– so because of that, we know that the

area of ABCD is just going to be equal to

2 times the area of, we could pick

either one of these. We could say 2 times

the area of ABC. Because area of ABCD– actually

let me write it this way. The area of ABCD is equal to

the area of ADC plus the area of ABC. But since they’re

congruent, these two are going to be the

same thing, so it’s just going to be 2 times

the area of ABC. Now what is the area of ABC? Well area of a triangle is

just 1/2 base times height. So area of ABC is just

equal to 1/2 times the base of that triangle times its

height, which is equal to 1/2. What is the length of the base? Well the length

of the base is AC. So it’s 1/2– I’ll

color code it. The base is AC. And then what is the height? What is the height here? Well we know that this

diagonal right over here, that it’s a

perpendicular bisector. So the height is just

the distance from BE. So it’s AC times BE,

that is the height. This is an altitude. It intersects this base

at a 90-degree angle. Or we could say BE is the

same thing as 1/2 times BD. So this is– let me write it. This is equal to, so it’s

equal to 1/2 times AC, that’s our base. And then our height is

BE, which we’re saying is the same thing

as 1/2 times BD. So that’s the area

of just ABC, that’s just the area of this broader

triangle right up there, or that larger triangle

right up there, that half of the rhombus. But we decided that the

area of the whole thing is two times that. So if we go back, if we

use both this information and this information right over

here, we have the area of ABCD is going to be equal to

2 times the area of ABC, where the area of ABC is

this thing right over here. So 2 times the area

of ABC, area of ABC is that right over there. So 1/2 times 1/2 is

1/4 times AC times BD. And then you see

where this is going. 2 times 1/4 fourth is

1/2 times AC times BD. Fairly straightforward,

which is a neat result. And actually, I haven’t

done this in a video. I’ll do it in the next video. There are other ways of finding

the areas of parallelograms, generally. It’s essentially base times

height, but for a rhombus we could do that because

it is a parallelogram, but we also have this

other neat little result that we proved in this video. That if we know the

lengths of the diagonals, the area of the rhombus

is 1/2 times the products of the lengths of the diagonals,

which is kind of a neat result.

I'm studying to be an RT, but, wow, I see all of your new posts. I'm envious of your plethera of knowledge!

Im only a 15 year old, and i love you man! subscribed!!

In what tool do you make these videos?

This was too easy and I like it

??????????????????

please upload videos on trapezium

SO BAD! DISLIKED

Anything on fourth dimensional objects or 3rd dimensional graps?

Mad

Very helpful for me