The length of a rectangle plus its width is 24 cm. the area is 143 square cm. what are the length and width of the rectangle?

The rectangle is 13cm by 11cm  (or 11cm by 13cm)

Step-by-step explanation:

Length plus width = 25

L+W = 24

Area is 143

A = LW

143 = LW

We have two equations and two unknowns

L+W = 24    Solve for W  by subtracting L from each side   W = 24-L

143 = LW

Now substitute into the second equation

143 = L(24-L)

Distribute

143 = 24L - L^2

Multiply by -1

-143 = -24L +L^2

-143 = L^2 -24L

Completing the square

-24/2 = -12   -12^2 =144

Add 144 to each side

-143+144 = L^2 -24L+144

1 =   (L-12)^2

Take the square root of each side

±sqrt(1) = sqrt(  (L-12)^2)

±sqrt(1) =   (L-12)

Add 12 to each side

12±sqrt(1) =   (L-12)+12

12±1 = L

13=L   or 11=L

Now we need to find W

W = 24 - L

W = 24-13   W = 24-11

W = 11 or W =13

First, illustrate the problem. 

Draw a lamp and a man standing in the middle of a stage, then draw a shadow of a man. Assign the lamp as one of the legs of the triangle and the other leg is the distance of the man from the lamp and the hypotenuse is the shadow of the man. Use the Pythagorean theorem and similar triangles.