Manhattan Prep Geo - #18 p. 92

[Alexander]

Hi -

"The lengths of two shorter legs of a right triangle add up to 40 units. What is the maximum possible area of the triangle"

I didn't think the explanation was very good or maybe I didnt see it quick enough. The explanation says, " you can think of a right triangle as half a rectangle" - shouldn't it say " half a rectangle or square?"

I ask because the explanation goes on to say " of all the rectangles, the square has the greatest area". Yes, I took note of this on p 86.

They key to this problem is making the connection that all squares are rectangles ( not vica versa), and that given a perimeter a square has the greatest area.

Also the explanation states that isosceles triangles have the greatest area of triangles. Manhattan Prep might consider including that on page 87 of the guide where it discusses maximizing triangles.

What do you think?

Thanks,
Alexander

[tommywallach]

Hey Alexander,

Two things:

1) The definition of a rectangle is any shape with 4 sides that all meet at ninety degree angles. So by definition, a square is a rectangle. So there is no need to say "rectangle or square." A square is a rectangle. In fact, it would be confusing to say that, because it would imply that they are separate entities.

2) I understand your point, but remember, the point of that book is to prepare you for the test. The point isn't that every chapter totally prepares you for every question in the back. Taken as a whole, the chapters AND the questions join together to teach you what you need to know. Also, it's worth noting that it should be logical that the best shape for a triangle is isosceles, because if you imagine a short but very long triangle extended as far as possible, it's very clear that the area is shrinking.

Hope that all makes sense! And thanks for all the posts. We love an active forum. : )

-t