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 $(\frac{-2}{11} ;\frac{3}{11})$.
[Math] coordinates of the vertices of the rhombus
coordinate systemsgeometry
Related Solutions
Let $B=(x,y)$ be the vertex below the given diagonal. Since $B$ is equidistant to $A$ and $C$: $$ (x-5)^2+(y-4)^2=(x+3)^2+(y+4)^2. $$ Simplifying the above yields $$\tag{1} 1=x+y $$
Since $BC$ has gradient $5/3$ $$ {y-4\over x-5}={5\over3}; $$ whence $$\tag{2}3y-5x=-13.$$
By $(1)$, we have $y=1-x$. Substituting into $(2)$ gives $ 3(1-x)-5x=-13$. This gives $x=2$; and, from $(1)$, $y=-1$.
So $B=(2,-1)$.
Since the given diagonal has slope 1, and since $B$ is 5 units to the right and 3 units up from $A$, the forth vertex is 5 units up and 3 to the right of $A$, So $D=(0,1 )$.
Hint: Here are the points on graph paper.
Best Answer
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 $5x+7y=1$)?