samichange wrote:

Does the rectangular mirror have an area greater than \(10 cm^2\) ?

1) The perimeter of the mirror is 24 cm.

2) The diagonal of the mirror is less than 11 cm.

Dear

samichange,

This is a great question! I'm happy to respond!

Statement #1:

Since a rectangle is a square, we could have a 6 x 6 square, that has an area of 36, more than 10. If we made this long and skinny, an 11 by 1 rectangle, that still has an area of 11, more than 10. But if we make it really skinny, 11.9 by 0.1, then it has an area of 1.19, much less than 10. This statement, alone and by itself, is

not sufficient.

Statement #2:

The 6 x 6 square discussed in the first statement has a diagonal of \(6sqrt(2)\), which is approximately 6*(1.4) = 8.4, which is less than 11. Again, this square has an area of 36, more than 10. Alternately, we could have a really tiny rectangle, 1 x 2, than has a diagonal much less than 11 and an area much less than 10. We could go either way. This statement, alone and by itself, is

not sufficient.

Combined.

Now, perimeter is fixed at 24, and the diagonal must be less than 11. Clearly, the square and rectangles in which length and width are close will have areas more than 10. As the rectangles of perimeter 24 get skinnier, the diagonals get longer and longer, and the areas go down.

Let's go back to the 11 x 1 triangle, which has an area of 11, greater than 10. This clearly has a diagonal greater than 11, so it wouldn't could under the combined statements. All the rectangles that are "less skinny", with a length/width ratio < 11, will have areas more than this, more than 11. Some of those rectangles will have diagonals less than 11, and so will count under the combined statements.

On the other hand, if we look at the rectangles that are "more skinny" than the 11 x 1 rectangle, with a length/width ratio > 11, then many of those will have areas less than 10, but all of those, like the 11 x 1 rectangle, will have diagonals larger than 11, so none of those would be included among the combined statements.

The only rectangles that would count are some subset of the rectangles that are "less skinny" than the 11 x 1, and we definitively can say that all of these have an area greater than 10. Combined, the statements are sufficient.

Does all this make sense?

Mike

_________________

Mike McGarry

Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)