Complete the explanation of how the figure illustrates that 6(9) = 6(4) + 6(5).

the area is the product of the length and width (6 × ? ). it is also the sum of the areas of the rectangles separated by the dished line (6 × ? and 6 × 5).
so, 6(9) = 6(4) + 6(5).

The explanation of how the figure illustrates that 6(9) = 6(4) + 6(5) is below.

Step-by-step explanation:

Consider a Rectangle ABCD segregate in two Rectangle by a Dash Line

i.e Rectangle AEFD  and

    Rectangle EBCF

We Know

For Rectangle ABCD we have

Length = 6

Width = 9

..........( 1 )

So For Rectangle AEFD we have

Length = 6

Width = 4

..........( 2 )

Similarly, For Rectangle EBCF we have

Length = 6

Width = 5

..........( 3 )

Now,

Substituting the values we get

Which is equal to

So, 6(9) = 6(4) + 6(5).

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4