Proof - Rhombus Area Half Product of Diagonal Length

Showing that we can find the area of a rhombus by taking half the product of the lengths of the diagonals

Quadrilateral ABCD they're telling us it is a rhombus

To prove that the area of this rhombus is equal to one half times x AC x BD,

essentially proving that the area of a rhombus is one half

times the product of the lengths of its diagonals

Let' s see what we can do over here

There's a bunch of things we know about rhombi

All rhombi are parallelograms and 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

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 that

the diagonals bisect each other

Let's call this point over here E

We know that BE is 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're 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

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

The first part is pretty straightforward

We know 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 AC right over here

So, this is by side-side-side

Because of that, we know that the area of ABCD is just going to be

equal to 2 times the area of, we can pick either one of these, ABC

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 2 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

The area of a triangle is just one half of base times height

The area of ABC is just equal to times the base of that triangle

times its height

What is the length of the base

The length of the base is AC

I'll color code it

The base is AC and then what is the height here

We know that this diagonal line over here is 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 can say BE is the same thing as times BD

This is equal to times AC, that's our base

Our height is BE, which is times BD

So that's the area of just ABC, that broader,

larger triangle right up there

That half of the rhombus

We just said that the area of the whole thing is 2 times that

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, this thing right over here

It is 2 times the area of ABC, right over there

So times is , times AC times BD

Then you see where this is going

2 times is , times AC times BD

Fairly straightforward, there's a neat result

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 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

And if we know that lengths of the diagonals,

the area of the rhombus is times the products of the lengths

of the diagonals, which is kind of a neat result