coordinates of the vertices of the rhombus

Question

One side of a rhombus lies along the line $5x +7y= 1$ and one of the vertices is $(3,-2)$. One diagonal of the rhombus is the line $3y=x + 1$.Can you find the coordinates of the other vertices and the equations of the three remaining sides? I have found one of the vertices $(\frac{-2}{11} ;\frac{3}{11})$.

Answer 1

HINT

Since you are looking for some other vertices, note that $(3,-2)$ is on the first line you mentioned. Diagonal of the rhombus is an axis of symmetry. Can you reflect the point $(-3,2)$ across the diagonal line to get another vertex (it should still lie on the same line $5x+7y=1$)?

Answer 2

Diagonals of rhombus are perpendicular so the equation for the other diagonal is $y=-3x+b$ and we know that vertex $(-2/11,3/11)$ belongs to it so $3/11=-6/11+b$, $b=9/11$. We have equation for two diagonals: $3y=x+1$ and $y=-3x+9/11$, let's find the point of intersection: $x+1=-9x+27/11$, $x=\frac{8}{55}, y=\frac{21}{55}$. Notice that this is a midpoint of each of the diagonal. Let's say the coordinates of the third vertex is $(x_3, y_3)$ then we have $(x_3+3)/2=\frac{8}{55}, x_3=-\frac{149}{55}$ and $(y_3-2)/2=\frac{21}{55}$, $y_3=\frac{141}{55}$. Similarly you can find the coordinates of the fourth vertex.

Answer 3

HINT

We know the diagonals of a rhombus are perpendicular. Do you know how to mirror a point on a given line? ( Straight line slope is negative reciprocal, passes through given point.)

Reflect a point D not on given diagonal to a point A. Next reflect one other given point on other side of new diagonal DA.