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Chapter 8: Q36. (page 415)

The quadrilateral RSTV has vertices $R (- 4, 6), S (4, 5), T (6, 3) and V (5, - 8)$ . Find the perimeter of quadrilateral.

Short Answer

Expert verified

The perimeter of quadrilateral is approximately $38.6 units$ .

Step by step solution

01

Step 1. Write down the given information.

The given quadrilateral RSTV has vertices $R (- 4, 6), S (4, 5), T (6, 3) and V (5, - 8)$ .

02

Step 2. Calculation.

The distance $(d)$ between the pair of points with given coordinates $R (- 4, 6), S (4, 5), T (6, 3) and V (5, - 8)$ is evaluated using the distance formula.

\begin{matrix} |d_{R S}| = \sqrt{{(x_{2} 鈭� x_{1})}^{2} + {(y_{2} 鈭� y_{1})}^{2}} \\ = \sqrt{{(4 + 4)}^{2} + {(5 鈭� 6)}^{2}} \\ = \sqrt{65} units \end{matrix}

\begin{matrix} |d_{S T}| = \sqrt{{(x_{2} 鈭� x_{1})}^{2} + {(y_{2} 鈭� y_{1})}^{2}} \\ = \sqrt{{(6 鈭� 4)}^{2} + {(3 鈭� 5)}^{2}} \\ = 2 \sqrt{2} units \end{matrix}

\begin{matrix} |d_{T V}| = \sqrt{{(x_{2} 鈭� x_{1})}^{2} + {(y_{2} 鈭� y_{1})}^{2}} \\ = \sqrt{{(5 鈭� 6)}^{2} + {(鈭� 8 鈭� 3)}^{2}} \\ = \sqrt{122} units \end{matrix}

\begin{matrix} |d_{R V}| = \sqrt{{(x_{2} 鈭� x_{1})}^{2} + {(y_{2} 鈭� y_{1})}^{2}} \\ = \sqrt{{(5 + 4)}^{2} + {(鈭� 8 鈭� 6)}^{2}} \\ = \sqrt{277} units \end{matrix}

Now the perimeter of quadrilateral is the sum of its four sides. Therefore perimeter (P) of quadrilateral (RSTV) is evaluated as:

\begin{matrix} P = |d_{R S}| + |d_{S T}| + |d_{T V}| + |d_{R V}| \\ = \sqrt{65} + 2 \sqrt{2} + \sqrt{122} + \sqrt{277} units \\ = 38.6 鈥塽苍颈迟蝉 (Approx.) \end{matrix}

03

Step 3. Conclusion.

The perimeter of quadrilateral (RSTV) is approximately $38.6 鈥� units$ .

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