Rectangle
A Rectangle is a four sided-shape where all the angles are right-angled (90o).
Properties
1. It’s a parallelogram with four right angles.
2. It’s diagonals bisect each other.
3. The opposite side of a rectangle is equal.
4. The opposite side of a rectangle is parallel.
The area and the perimeter of Rectangle
The area of a Rectangle is the plan enclosed by the sides of the rectangle.
The area is found by multiplying the length with the width.
Area = length x width
$ \displaystyle A=a\cdot b$
The perimeter of a Rectangle is the total lengths of all sides.
The perimeter is 2 times the (length of one side +the other side)
$ \displaystyle P=2\left( a+b \right)$
Example 1: On the figure below the sides of rectangle are $ \displaystyle AC=25cm$ and $ \displaystyle BC=7cm$. Find the area.
Solution: The diagonal separates the rectangle into two equal right angle triangle.
On triangle $ \displaystyle ABC$ we find $ \displaystyle AB$ by using the Pythagorean theorem
$ \displaystyle {{\left( {AB} \right)}^{2}}={{\left( {AC} \right)}^{2}}-{{\left( {BC} \right)}^{2}}$
$ \displaystyle {{\left( {AB} \right)}^{2}}={{\left( {25} \right)}^{2}}-{{\left( 7 \right)}^{2}}$
$ \displaystyle {{\left( {AB} \right)}^{2}}=625-49$
$ \displaystyle {{\left( {AB} \right)}^{2}}=567$
$ \displaystyle AB=24cm$
Now that we know the width and length of the rectangle we find the area.
$ \displaystyle Area=AB\cdot BC=24\cdot 7=168c{{m}^{2}}$
Example 2: On the figure below the sides of the rectangle are$ \displaystyle AD=3cm$ and $ \displaystyle DC=7cm$. The segment $ \displaystyle CE=5cm$. Find the area of the triangle $ \displaystyle ACE$?
Solution: Based on the property that opposite sides of rectangle are equal $ \displaystyle AD=BC=3cm$
We observe the triangle $ \displaystyle EBC$ that is a right angle triangle because of the right angle of the rectangle.
We find $ \displaystyle EB$ by using the Pythagorean theorem
$ \displaystyle {{\left( {EB} \right)}^{2}}={{\left( {EC} \right)}^{2}}-{{\left( {CB} \right)}^{2}}$
$ \displaystyle {{\left( {EB} \right)}^{2}}={{\left( 5 \right)}^{2}}-{{\left( 3 \right)}^{2}}$
$ \displaystyle {{\left( {EB} \right)}^{2}}=25-9$
$ \displaystyle {{\left( {EB} \right)}^{2}}=16$
$ \displaystyle EB=4$
To find the area of the triangle $ \displaystyle ACE$ we can find the area of the rectangle and subtract the area of the two right angle triangles $ \displaystyle ADC$ and $ \displaystyle EBC$
$\displaystyle {{A}_{{ABCD}}}=AD\cdot DC=$$\displaystyle 7\cdot 3=21c{{m}^{2}}$
$\displaystyle {{A}_{{ADC}}}=$$\displaystyle \frac{1}{2}AD\cdot DC=$$\displaystyle \frac{{21}}{2}c{{m}^{2}}$
$\displaystyle {{A}_{{EBC}}}=\frac{1}{2}EB\cdot BC=$$\displaystyle \frac{{3\cdot 4}}{2}=6c{{m}^{2}}$
$\displaystyle {{A}_{{AEC}}}=$$ \displaystyle {{A}_{{ABCD}}}-({{A}_{{ADC}}}+{{A}_{{EBC}}})$
$\displaystyle {{A}_{{AEC}}}=$$\displaystyle 21-(\frac{{21}}{2}+6)=$$\displaystyle \frac{{42}}{2}-\frac{{33}}{2}=\frac{9}{2}c{{m}^{2}}$
So the area of the triangle is $ \displaystyle {{A}_{{AEC}}}=\frac{9}{2}=4,5c{{m}^{2}}$
Example 3: The perimeter of a rectangle is $ \displaystyle 150cm$. If the widthis $ \displaystyle 30cm$, find the length and area?
Solution: We know that the perimeter is found by adding the length of all sides.
Lets note one side with $ \displaystyle x$.
From the properties the opposite sides of a rectangle are congruent so:
$ \displaystyle P=x+x+30+30=150$
$ \displaystyle 2x+60=150$
$ \displaystyle 2x=90$
$ \displaystyle x=45$
So the length is $ \displaystyle 45cm$.
The area is found by multiplying the length and width.
So $\displaystyle Area=45\cdot 30=1350c{{m}^{2}}$
Example 4: On the figure below we know that the diagonals are $ \displaystyle 16cm$. Find x and y.
Solution: From the properties of rectangle we know that diagonals bisect each other and are congruent.
$ \displaystyle AC=16$ so $ \displaystyle AO=OC=\frac{{16}}{2}=8$
$ \displaystyle AO=x+y=8$ and $ \displaystyle OC=3x-y=8$
Now to find $ \displaystyle x$ and $ \displaystyle y$ we have to solve this Simultaneous Linear Equations with two variables.
Solving simultaneously using Elemination Method
$ \displaystyle \left\{ \begin{array}{l}x+y=8\\3x-y=8\end{array} \right.$
$ \displaystyle \frac{\begin{array}{l}x+y=8\\3x-y=8\end{array}}{{4x=16}}$
$ \displaystyle x=4$
Now we substitute to find the other variable
$ \displaystyle 4+y=8$
$ \displaystyle y=4$
So $ \displaystyle x=4$ and $ \displaystyle y=4$
Example 5: Find the length and width of the rectangle if it´s area is $ \displaystyle 5c{{m}^{2}}$.
Solution: We know that the area of the rectangle is width x length
$ \displaystyle Area=(x+3)(x-1)=5$
$ \displaystyle (x+3)(x-1)=5$
$ \displaystyle {{x}^{2}}-x+3x-3=5$
$ \displaystyle {{x}^{2}}+2x-3-5=0$
$ \displaystyle {{x}^{2}}+2x-8=0$
TIP! To solve quadratic equation check Quadratic equation
$ \displaystyle {{x}_{1}}=2$ and $ \displaystyle {{x}_{2}}=-4$
Since we are talking about width and length only $ \displaystyle {{x}_{1}}=2$ is acceptable.
Finding the length: $ \displaystyle x+3=2+3=5cm$
Finding the width: $ \displaystyle x-1=2-1=1cm$
The proof: $ \displaystyle A=5\cdot 1=5c{{m}^{2}}$
