the length of a triangle is 7 centimeters less than twice the width. if the area of the rectangle is 165 square centimeters, find the length and width.

L = 2W-7

L * W = 165

Substitute 2W-7 for L in the second equation and solve for W. Insert that value into the first equation to solve for L. Check by putting both values into the second equation.

To solve this problem, we need to set up equations using the given information.

Let's assume the width of the rectangle is represented by 'w' cm.
According to the problem, the length of the rectangle is 7 centimeters less than twice the width.
So, the length can be expressed as 2w - 7 cm.

To find the area of the rectangle, we multiply the length by the width:
Area = Length * Width

In this case, we have the area given as 165 square centimeters:
165 = (2w - 7) * w

Now we can solve for 'w' by simplifying the equation and solving for the quadratic form.

165 = 2w^2 - 7w

Rearranging the equation:
2w^2 - 7w - 165 = 0

We can now factor or use the quadratic formula to solve for 'w'. Let's use the quadratic formula to find the two possible values for 'w':

w = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this equation, a = 2, b = -7, and c = -165.

w = (-(-7) ± sqrt((-7)^2 - 4 * 2 * -165)) / (2 * 2)
w = (7 ± sqrt(49 + 1320)) / 4
w = (7 ± sqrt(1369)) / 4
w = (7 ± 37) / 4

Now we have two possible values for 'w':
w1 = (7 + 37) / 4 = 44 / 4 = 11 cm
w2 = (7 - 37) / 4 = -30 / 4 = -7.5 cm (We discard the negative value since width cannot be negative in this context.)

Therefore, the width of the rectangle is 11 cm.

To find the length, substitute the value of 'w' back into the expression we found earlier for the length:
Length = 2w - 7
Length = 2 * 11 - 7
Length = 22 - 7
Length = 15 cm

So, the length of the rectangle is 15 cm and the width is 11 cm.