Summarize the following:
You can isolate the length (l{\displaystyle l}) or the width (w{\displaystyle w}). Set this formula aside. You will plug it into the area formula later. For example, if you know the width of a rectangle is 2 cm more than the length, you can write a formula for w{\displaystyle w}: w=l+2{\displaystyle w=l+2}. The formula is A=lw{\displaystyle A=lw}, where A{\displaystyle A} equals the area of the rectangle, l{\displaystyle l} equals the length of the rectangle, and w{\displaystyle w} equals the width of the rectangle. You can use this method if you know the perimeter of the rectangle, except you would now set up the perimeter formula instead of the area formula. The formula for the perimeter of a rectangle is P=2(w+l){\displaystyle P=2(w+l)}, where w{\displaystyle w} equals the width of the rectangle, and l{\displaystyle l} equals the length of the rectangle. Make sure you substitute for the variable A{\displaystyle A}. For example, if the area of the rectangle is 35 square centimeters, your formula will look like this: 35=lw{\displaystyle 35=lw}. Since you are working with a rectangle, it doesn't matter whether you work with the l{\displaystyle l} or w{\displaystyle w} variable. For example, if you found that w=l+2{\displaystyle w=l+2}, then you would substitute this relationship for w{\displaystyle w} in the area formula:35=lw{\displaystyle 35=lw}35=l(l+2){\displaystyle 35=l(l+2)} To do this, use the distributive property to multiply the terms in parentheses, then set the equation to 0. For example:35=l(l+2){\displaystyle 35=l(l+2)}35=l2+2l{\displaystyle 35=l^{2}+2l}0=l2+2l35{\displaystyle 0=l^{2}+2l-35} For complete instructions on how to do this, read Solve Quadratic Equations. For example, the equation 0=l2+2l35{\displaystyle 0=l^{2}+2l-35} can be factored as 0=(l+7)(l5){\displaystyle 0=(l+7)(l-5)}. To do this, set each term to zero and solve for the variable. You will find two solutions, or roots, to the equation. For example:0=(l+7){\displaystyle 0=(l+7)}7=l{\displaystyle -7=l}AND0=(l5){\displaystyle 0=(l-5)}5=l{\displaystyle 5=l}.In this case, you have one negative root. Since the length of a rectangle cannot be negative, you know that the length must be 5 cm. This will give you the length of the other side of the rectangle. For example, if you know that the length of the rectangle is 5 cm, and that the relationship between the side lengths is w=l+2{\displaystyle w=l+2}, you would substitute 5 for the length in the formula:w=l+2{\displaystyle w=l+2}w=5+2{\displaystyle w=5+2}w=7{\displaystyle w=7} The formula is a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}}, where a{\displaystyle a} and b{\displaystyle b} equal the side lengths of a right triangle, and c{\displaystyle c} equals the length of a right triangle's hypotenuse. You use the Pythagorean Theorem because a diagonal of a rectangle cuts the rectangle into two congruent right triangles. The width and length of the rectangle are the side lengths of the triangle; the diagonal is the hypotenuse of the triangle. It doesn't matter which value you use for which variable. For example, if you found the width and length of the rectangle are 5 cm and 7 cm, your formula will look like this: 52+72=c2{\displaystyle 5^{2}+7^{2}=c^{2}}. Remember, squaring a number means to multiply the number by itself. For example:52+72=c2{\displaystyle 5^{2}+7^{2}=c^{2}}25+49=c2{\displaystyle 25+49=c^{2}}74=c2{\displaystyle 74=c^{2}} The easiest way to find a square root is to use a calculator. You can use an online calculator if you do not have a scientific calculator. This will give you the value of c{\displaystyle c}, which is the hypotenuse of the triangle, and the diagonal of the rectangle. For example:74=c2{\displaystyle 74=c^{2}}74=c2{\displaystyle {\sqrt {74}}={\sqrt {c^{2}}}}8.6024=c{\displaystyle 8.6024=c}So, the diagonal of a rectangle with a width that is 2 cm more than the length, and an area of 35 cm, is about 8.6 cm.
Write a formula explaining the relationship between the side lengths. Set up the formula for the area of a rectangle. Plug the area of the rectangle into the formula. Plug the relational formula for the length (or width) into the formula. Set up a quadratic equation. Factor the quadratic equation. Find the values of l{\displaystyle l}. Plug the value of the length (or width) into your relationship formula. Set up the formula for the Pythagorean Theorem. Plug the width and length into the formula. Square the width and length, then add these numbers together. Take the square root of each side of the equation.