Question

A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 24 m. Each vertex of the rhombus is a midpoint of the side of the rectangle. Express the area of the rhombus as a function of the rectangle's width.

Answer 1

I would encourage you to have a visual (draw a diagram) to begin.

Answer 2

OH GOOD!

Answer 3

Let L represent the rectangles length.

Answer 4

Now w and L are the width and length respectively. Do you notice there are four congruent right triangles?

Answer 5

The vertices of the rhombus are located at the midpoints along the width and the length of the rectangle, corect?

Answer 6

So first, what is the equation for the perimeter of the rectangle, in terms of w and L?

Answer 7

As a teacher, I make time to come here and teach, and thus am NOT going to just give you the answer so a little participation from you would be greatly appreciated.

Answer 8

NO?? Well then good luck!

Answer 9

I'll return when I see you're ready!

Answer 10

Give me a second.

Answer 11

@calculusfunctions P=2w+2L

Answer 12

I do not have the best the connection on here so I may be slow to respond..

Answer 13

Correct! However since the perimeter is 24 m, then 24 = 2L + 2w OR L + w = 12. Now concentrate on one the four congruent right triangles, and tell me the expression for each side of the rhombus in terms of L and w.

Answer 14

I am a bit lost.

Answer 15

Actually, never mind. The area of a rhombus is\[A =\frac{ d _{1}d _{2} }{ 2 }\]where the two d's are the diagonals of the rhombus. Now use this hint to answer the question.

Answer 16

I am sorry, but I am completely lost on what I am supposed to do next.

Answer 17

What are the lengths of the diagonals of the rhombus in terms of L and w?

Answer 18

Look at the diagram Carefully. What the lengths of the diagonals af the rhombus? Do you know what a diagonal is?

Answer 19

Earlier with the triangle thing did you mean L=w-12and W=L-12

Answer 20

But then I said "never mind". Don't worry about the triangles.

Answer 21

And yes I think I do know which the diagonals are

Answer 22

So then you would see that the diagonals of the rhombus are parallel to the the length and width of the rectangle. Hence the diagonals are L and w. Understand?

Answer 23

Yes I understand that.

Answer 24

Thus the area of the rhombus is\[A =\frac{ Lw }{ 2 }\]However you were asked to express the area in terms of w. Thus from the simplified equation (L + w = 12) for the perimeter of the rectangle, from earlier, solve for L and substitute for L into the above area equation.

Answer 25

Tell me what you get.

Answer 26

So when you solve for L you get L=12-w and when you substitute it you get A=w(12-w)/2

Answer 27

Yes! Now simplify.

Answer 28

I wait isn't \[\frac{ w(w+12) }{ 2 }\] already simplified? or would it be\[\frac{ 1 }{ 2 }w(w+12)\]

Answer 29

NO! You mean\[A =\frac{ 1 }{ 2 }w(12-w)\]Now simplify.

Answer 30

Use the distributive property to simplify.

Answer 31

\[A=6w-\frac{ 1 }{ 2 }w^2\]

Answer 32

Good! That's correct!

Answer 33

Thank you for helping!

Answer 34

Welcome! Anything else?

Answer 35

I think I'm good for now.