You use the Pythagorean Theorem because a diagonal of a rectangle cuts the rectangle into two congruent right triangles. [4] X Research source The length and width of the rectangle are the side lengths of the triangle; the diagonal is the hypotenuse of the triangle.
For example, if the width of a rectangle is 3 cm, and the length is 4 cm, your formula will look like this: 32+42=c2{\displaystyle 3^{2}+4^{2}=c^{2}}.
For example:32+42=c2{\displaystyle 3^{2}+4^{2}=c^{2}}9+16=c2{\displaystyle 9+16=c^{2}}25=c2{\displaystyle 25=c^{2}}
For example:25=c2{\displaystyle 25=c^{2}}25=c2{\displaystyle {\sqrt {25}}={\sqrt {c^{2}}}}5=c{\displaystyle 5=c}So, the diagonal of a rectangle with a width of 3 cm and a length of 4 cm is 5 cm.
For example, if the area of the rectangle is 35 square centimeters, your formula will look like this: 35=lw{\displaystyle 35=lw}.
For example:35=lw{\displaystyle 35=lw}35l=w{\displaystyle {\frac {35}{l}}=w}.
For example, if the perimeter of a rectangle is 24 centimeters, your formula will look like this: 24=2(w+l){\displaystyle 24=2(w+l)}.
For example:24=2(w+l){\displaystyle 24=2(w+l)}242=2(w+l)2{\displaystyle {\frac {24}{2}}={\frac {2(w+l)}{2}}}12=w+l{\displaystyle 12=w+l}.
For example, if using the area formula you found that 35l=w{\displaystyle {\frac {35}{l}}=w}, replace this value of w{\displaystyle w} into the perimeter formula:12=w+l{\displaystyle 12=w+l}12=35l+l{\displaystyle 12={\frac {35}{l}}+l}
For example:12=35l+l{\displaystyle 12={\frac {35}{l}}+l}12×l=(35l×l)+(l×l){\displaystyle 12\times l=({\frac {35}{l}}\times l)+(l\times l)}12l=35+l2{\displaystyle 12l=35+l^{2}}
For example:12l=35+l2{\displaystyle 12l=35+l^{2}}12l−12l=35+l2−12l{\displaystyle 12l-12l=35+l^{2}-12l}0=35+l2−12l{\displaystyle 0=35+l^{2}-12l}
For example, 0=35+l2−12l{\displaystyle 0=35+l^{2}-12l} becomes 0=l2−12l+35{\displaystyle 0=l^{2}-12l+35}.
For example, the equation 0=l2−12l+35{\displaystyle 0=l^{2}-12l+35} can be factored as 0=(l−7)(l−5){\displaystyle 0=(l-7)(l-5)}.
For example:0=(l−7){\displaystyle 0=(l-7)}7=l{\displaystyle 7=l}AND0=(l−5){\displaystyle 0=(l-5)}5=l{\displaystyle 5=l}. So, the length and width of the rectangle are 7 cm and 5 cm.
You use the Pythagorean Theorem because a diagonal of a rectangle cuts the rectangle into two congruent right triangles. [12] X Research source The width and length of the rectangle are the side lengths of the triangle; the diagonal is the hypotenuse of the triangle.
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}}.
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}}
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 an area of 35 cm and a perimeter of 24 cm is about 8. 6 cm.
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}.
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. [16] X Research source
For example, if the area of the rectangle is 35 square centimeters, your formula will look like this: 35=lw{\displaystyle 35=lw}.
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)}
For example:35=l(l+2){\displaystyle 35=l(l+2)}35=l2+2l{\displaystyle 35=l^{2}+2l}0=l2+2l−35{\displaystyle 0=l^{2}+2l-35}
For example, the equation 0=l2+2l−35{\displaystyle 0=l^{2}+2l-35} can be factored as 0=(l+7)(l−5){\displaystyle 0=(l+7)(l-5)}.
For example:0=(l+7){\displaystyle 0=(l+7)}−7=l{\displaystyle -7=l}AND0=(l−5){\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.
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}
You use the Pythagorean Theorem because a diagonal of a rectangle cuts the rectangle into two congruent right triangles. [19] X Research source The width and length of the rectangle are the side lengths of the triangle; the diagonal is the hypotenuse of the triangle.
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}}.
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}}
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.
