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)=√((-4-1)^2+(5-4)^2)`

`=√((-5)^2+(1)^2)`

`=√(25+1)`

`=√(26)`

`d_(MB)=√((1-2)^2+(4+1)^2)`

`=√((-1)^2+(15)^2)`

`=√(1+25)`

`=√(26)`

`d_(BS)=√((2+3)^2+(-1-0)^2)`

`=√((5)^2+(-1)^2)`

`=√(25+1)`

`=√(26)`

`d_(SR)=√((-3+4)^2+(0-5)^2)`

`=√((1)^2+(-5)^2)`

`=√(1+25)`

`=√(26)`

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

Diagonals:

Use the slope formula:
`m=(y_2-y_1)/(x_2-x_1)`

`m_(RB)=(5+1)/(-4-2)`

`=(6)/(-6)`

= -1

`m_(MS)=(4-0)/(1+3)`

`=(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.