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I NEED HELP ON THIS ASAP!!!!

Horace says that he determined the area of Rectangle ABCD by determining the product (CD) (CB). Bernice says that Horace is incorrect because he needs to use the base of the rectangle and that the base is AB, not CD. Horace responded by saying that CD is one of the bases. Who’s correct? Explain your reasoning.

I NEED HELP ON THIS ASAP!!!! Horace says that he determined the area of Rectangle-example-1

1 Answer

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Answer:

Horace

Explanation:

You want to know if "base" refers to any particular side of a rectangle.

Area formula

The formula for the area of a rectangle can be written ...

Area = base × height

It can also be written ...

Area = length × width

In each formula, the factors of the area refer to orthogonal dimensions of the rectangle. Any side of the rectangle can be considered to be the "base" or the "length" or the "width" or the "height". The other factor in the area formula is whatever dimension is orthogonal to that choice.

Properties of arithmetic

The commutative property of multiplication tells you that you get the same product if you reverse the order of the numbers.

So, you could have ...

Area = AB·BC

or

Area = BC·AB

If you insist that the fist of these numbers is the "base", then the base can be either AB or BC and the area will be the same.

The substitution property of equality tells you that you can substitute any equal values. Opposite sides of a rectangle are equal length, so either can be substituted for the other. In short, you can write the product for the area any of 8 ways:

Area = AB·BC = AB·AD = BC·AB = AD·AB

= CD·AD = CD·BC = AD·CD = BC·CD

If you want to distinguish between AB and BA, you can write this many more ways.

Horace seems to understand the fundamental interchangeability of the dimensions in the area formula. Horace has the right idea.

Bernice seems more attached to one particular understanding. We wonder what Bernice would call the "base" if the rectangle were drawn at a different angle to the grid.

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