Question

The coordinates of the vertices of parallelogram RMBS are R(–4, 5), M(1, 4), B(2, –1), and S(–3, 0). Using the diagonals, prove that RMBS is a rhombus. Show all your work and state appropriate formulas and theorems used.

Answer 1

To prove a rhombus you need to show that the sides are congruent and the diagonals are perpendicular.

Sides:

$d_{RM}=\sqrt{(-4-1)^2+(5-4)^2}$

    $=\sqrt{(-5)^2+(1)^2}$

    $=\sqrt{25+1}$

    $=\sqrt{26}$

$d_{MB}=\sqrt{(1-2)^2+(4+1)^2}$

    $=\sqrt{(-1)^2+(15)^2}$

    $=\sqrt{1+25}$

    $=\sqrt{26}$

$d_{BS}=\sqrt{(2+3)^2+(-1-0)^2}$

    $=\sqrt{(5)^2+(-1)^2}$

    $=\sqrt{25+1}$

    $=\sqrt{26}$

$d_{SR}=\sqrt{(-3+4)^2+(0-5)^2}$

    $=\sqrt{(1)^2+(-5)^2}$

    $=\sqrt{1+25}$

    $=\sqrt{26}$

$\overline{RM}$ ≅ $\overline{MB}$ ≅ $\overline{BS}$ ≅ $\overline{SR}$

Diagonals:

Use the slope formula: $m=\dfrac{y_2-y_1}{x_2-x_1}$

$m_{RB}=\dfrac{5+1}{-4-2}$

       $=\dfrac{6}{-6}$

       = -1

$m_{MS}=\dfrac{4-0}{1+3}$

       $=\dfrac{4}{4}$

       = 1

Slopes are opposite reciprocals so they are perpendicular.

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All of the sides are congruent and the diagonals are perpendicular so RMBS is a rhombus.