What is the length of the diagonal?

Question

The width of a rectangle is 4 less than twice its length. If the area of the rectangle is 39cm^2, what is the length of the diagonal?

Answer 1

When we draw a diagonal across a rectangle, we get two congruent right triangles.  First, we need to find the length and width of the rectangle.  Then apply Pythagorean theorem to find the length of the diagonal.

 

Let length = x

Let width = 2x - 4

Area = 39 cm²

 

 

Set the equation to represent the area of a rectangle using the variables.

 

x(2x - 4) = 39

2x² - 4x = 39

2x² - 4x - 39 = 0

 

 

We have a quadratic equation.  Use the quadratic formula to solve for x.

 

x = (4 ± √(16 - 4(-78))) / 4

x = (4 ± √(328)) / 4

x = (4 ± 18.11) / 4

x = (4 - 18.11) / 4                and            x = (4 + 18.11) / 4

 

x = -3.53                            and             x = 5.53

 

 

Since we cannot have a negative value for dimensions, we accept the positive value of x:

 

x = 5.53

 

 

Plug in this value of x into the dimensions.

 

length = 5.53 cm

width = 7.06 cm

 

These dimensions are also the legs of the triangle that is formed by the diagonal.

 

 

Now using the Pythagorean theorem to find the length of the diagonal

 

diagonal = √(5.53² + 7.06²)

 

 

I will let you finish from here.