To solve this problem, we need to find the value of [tex]\( k \)[/tex] in the equation of the diagonal of the parallelogram, given as [tex]\( 3y = 5x + k \)[/tex]. The two opposite vertices of the parallelogram are given as [tex]\((1, -2)\)[/tex] and [tex]\((-2, 1)\)[/tex].
Let's proceed with the solution step-by-step:
1. Determine the Midpoint of the Diagonal:
The midpoint of the diagonal of a parallelogram (joining two opposite vertices) can be found using the midpoint formula, which states that the coordinates of the midpoint [tex]\((M_x, M_y)\)[/tex] between points [tex]\((x1, y1)\)[/tex] and [tex]\((x2, y2)\)[/tex] are:
[tex]\[
M_x = \frac{x1 + x2}{2}, \quad M_y = \frac{y1 + y2}{2}
\][/tex]
Substituting the given points [tex]\((1, -2)\)[/tex] and [tex]\((-2, 1)\)[/tex]:
[tex]\[
M_x = \frac{1 + (-2)}{2} = \frac{-1}{2} = -0.5
\][/tex]
[tex]\[
M_y = \frac{-2 + 1}{2} = \frac{-1}{2} = -0.5
\][/tex]
So, the midpoint of the diagonal is [tex]\((-0.5, -0.5)\)[/tex].
2. Substitute the Midpoint into the Line Equation to Solve for [tex]\( k \)[/tex]:
The midpoint [tex]\((-0.5, -0.5)\)[/tex] should lie on the line described by the equation [tex]\(3y = 5x + k\)[/tex]. Substituting [tex]\(M_x = -0.5\)[/tex] and [tex]\(M_y = -0.5\)[/tex] into the line equation:
[tex]\[
3(-0.5) = 5(-0.5) + k
\][/tex]
Simplify and solve for [tex]\( k \)[/tex]:
[tex]\[
-1.5 = -2.5 + k
\][/tex]
Add [tex]\(2.5\)[/tex] to both sides of the equation:
[tex]\[
k = -1.5 + 2.5 = 1
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( 1 \)[/tex].
