Question

A rhombus, COAL is centered at the origin. The longer diagonal is on the y-axis and has a

length of 27m. The shorter diagonal is on the x-axis and has a length of has diagonals with

length 19m. Determine the coordinates of the vertices.

Answer 1

Answer with Step-by-step explanation:

We are given that

COAL is a rhombus .

CA and LO are the diagonals of rhombus COAL.

CA=27 m

OL=19 m

We know that

Diagonals of rhombus are bisect to each other.

Let P(0,0) be the intersecting point of diagonals

Therefore, CP=PA

LP=PO

Length PO=
`(1)/(2)LO=(19)/(2)=9.5 m`

Length of CP=
`(1)/(2)PA=\farc{1}{2}(27)=13.5 m`

Let C(
`0,y_2),O(x_1,0),A(0,y_1) \;and\;L(x_2,0)`.

Distance formula:
`√((x_2-x_1)^2+(y_2-y_1)^2)`

Using the formula

`CP=√((0-0)^2+(0-y_2)^2)=13.5`

`√(y^2_2)=13.5`

`y_2=\pm 13.5`

C(0,13.5) or (0,-13.5)

`√((y_2-y_1)^2)=27`

Substitute the value
`y_2=13.5`

`√(13.5-y_1)=27`

`13.5-y_1=\pm 27`

`y_1=13.5-27=-13.5`

A(0,-13.5)

`13.5-y_1=-27`

`y_1=13.5+27=40.5`

A(0,40.5)

Substitute
`y_2=-13.5`

`√((-13.5-y_1)^2)=27`

`-13.5-y_1=\pm 27`

`-13.5-y_1=27`

`y_1=-13.5-27=-40.5`

A(0,-40.5)

`-13.5-y_1=-27`

`y_1=-13.5+27=13.5`

A(0,13.5)

`LP=√((x_2-0)^2)=9.5`

`√(x^2_2)=9.5`

`x_2=\pm 9.5`

L(9.5,0) or L(-9.5,0)

`√(x_2-x_1)^2)=19`

`(9.5-x_1)^2}=19`

`9.5-x_1=\pm 19`

`9.5-x_1=19`

`x_1=9.5-19=-9.5`

`9.5-x_1=-19`

`x_1=9.5+19=28.5`

`(-9.5-x_1)=19`

`-9.5-x_1=\pm 19`

`-9.5-x_1=19`

`x_1=-9.5-19=-28.5`

`-9.5-x_1=-19`

`-9.5+19=x_1`

`x_1=9.5`

O(-9.5,0) or O(28.5,0) or (-28.5,0) or (9.5,0)