Arectangle has an area of 12 square centimeters and a perimeter of 16 centimeters. which of the following could be its dimensions?
a: 2 cm and 6 cm
b: 3 cm and 4 cm
c: 1.5 cm and 8 cm
d: 1 cm and 12 cm

Option A 2 cm and 6 cm

Step-by-step explanation:

A rectangle has an area of 12 square centimeters and a perimeter of 16 cm.

We have to find the dimensions of the rectangle.

Let the length of perimeter = x and width of y cm

so area = x × y = 12 (1)

and perimeter = 2 (x+y) = 16

                          x + y = 8 (2)

Now x = (8-y)

So from equation (1)

y(8-y) = 12

8y - y² = 12

y² - 8y + 12 = 0

y² - 6y - 2y + 12 = 0

y(y-6) -2 ( y - 6 ) = 0

( y-2 ) ( y-6 ) = 0 ⇒ y = z or y = 6

Now we put the values of y in equation (1)

                     x × y = 12

for y = 2        2x = 12

                      x = 6

for y = 6        6x = 12

                      x = 2

Therefore, possible dimensions may be 2 and 6 cm.

Option A 2 cm and 6 cm

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4