Geometry Problems of the Day

The following problems were taken from the GEOMETRY (COMMON CORE) Regents Exam given on Thursday, January 26, 2017.
Previous problems can be found here.

Part 1

7. The diagram below shows two similar triangles.

If tan O = 3/7, what is the value of x, to the nearest tenth?

(2) 5.6.
tan O = 3/7, and tan O = 2.4/x, so 3/7 = 2.4/x.
3x = (7)(2.4) = 16.8
x = 5.6

8. A farmer has 64 feet of fence to enclose a rectangular vegetable garden. Which dimensions would result in the biggest area for this garden?

(1) the length and the width are equal
A dilation increased the size, so it will no longer be congruent.
For all rectangles with the same perimeter, a square will always have the greatest area.
Consider this:
A square with sides equal to x, will have a perimeter of 4x and area of x².
A rectangle with sides equal to (x+1) and (x-1), will have a perimeter of 4x and area of x² - 1.
A rectangle with sides equal to (x+2) and (x-2), will have a perimeter of 4x and area of x² - 4.
A rectangle with sides equal to (x+3) and (x-3), will have a perimeter of 4x and area of x² - 9.
etc.
The area will always be some perfect subtracted from x², which will always be less than x².




Continue to the next problems.