# Sara used 34 meters of fencing to enclose a rectangular region. To be sure that the region was a rectangle, she measured the diagonals and found that they were 13 meters each. What are the length and width of the rectangle?

Jan 18, 2018

Length(L) $= 4$ meters

Width(W) $= 13$ meters

#### Explanation:

Given:

Sara used $34$ meters of fencing to enclose a rectangular region.

Hence,

Perimeter of a rectangle as shown below is $34$ meters

Hence 2x(Length + Width) = 34 meters

Let us assume that Length = L meters and Width = W meters.

So, $2 \cdot \left(L + W\right) = 34$ meters

What is below is a rough sketch and NOT drawn to scale

Hence,

AB = CD = L meters

AC = BD = W meters

We are given that Diagonals are 13 meters long

We know that,

the diagonals of a rectangle are equal length;

diagonals of a rectangle also bisect each other

What is below is a rough sketch and NOT drawn to scale

Angle $\angle A C D$ is right-angle

Using Pythagoras Theorem, we can write

$A {C}^{2} + C {D}^{2} = A {D}^{2}$

$\Rightarrow {W}^{2} + {L}^{2} = {13}^{2}$

$\Rightarrow {W}^{2} + {L}^{2} = 169$

Add $- {W}^{2}$ on both sides to get

$\Rightarrow {W}^{2} + {L}^{2} - {W}^{2} = 169 - {W}^{2}$

$\Rightarrow \cancel{{W}^{2}} + {L}^{2} - \cancel{{W}^{2}} = 169 - {W}^{2}$

$\Rightarrow {L}^{2} = 169 - {W}^{2}$

Taking square root on both sides

$\Rightarrow \sqrt{{L}^{2}} = \sqrt{169 - {W}^{2}}$

$\Rightarrow L = \pm \sqrt{{13}^{2} - {W}^{2}}$

We consider only positive values

$\Rightarrow L = \sqrt{{13}^{2}} - \sqrt{{W}^{2}}$

$\Rightarrow L = 13 - W$

Substitute $\textcolor{red}{L = \left\{13 - W\right\}}$ in $\textcolor{b l u e}{\left\{{W}^{2} + {L}^{2}\right\} = 169}$

$\Rightarrow {W}^{2} + {\left(13 - W\right)}^{2} = 169$

Using the identity $\textcolor{g r e e n}{{\left(a - b\right)}^{2} \equiv {a}^{2} - 2 a b + {b}^{2}}$ we get

$\Rightarrow {W}^{2} + 169 - 26 W + {W}^{2} = 169$

$\Rightarrow {W}^{2} + \cancel{169} - 26 W + {W}^{2} = \cancel{169}$

$\Rightarrow 2 {W}^{2} - 26 W = 0$

$\Rightarrow 2 W \left(W - 1 3\right) = 0$

$\Rightarrow W - 13 = 0$

Hence, $W = 13$

Hence, width of the rectangle = $13$ meters

$2 \cdot \left(L + W\right) = 34$ meters

Substitute the value of $W = 13$ to get

$2 \cdot \left(L + 13\right) = 34$

$\Rightarrow 2 L + 26 = 34$

Add $- 26$ to both sides

$\Rightarrow 2 L + \cancel{26} - \cancel{26} = 34 - 26$

$\Rightarrow 2 L = 34 - 26 = 8$

$\Rightarrow 2 L = 8$

$L = \frac{8}{2} = 4$

Length of the rectangle = 4 meters

Substitute the values of $L = 4 \mathmr{and} W = 13$ in

$2 \cdot \left(L + W\right) = 34$ meters

to verify our results

We get

$2 \cdot \left(4 + 13\right) = 34$ meters

$\Rightarrow 34 = 34$

Hence, our rectangle has

Length(L) $= 4$ meters

Width(W) $= 13$ meters