let mid – point of sides of Δ are  (5/2,3), (5/2,7) & (4,5) If incentre is (h, k) then value of 3h + k is

The correct answer is 13.

Let the vertices of the triangle be $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$.

We are given the mid-points of the sides of the triangle as:
$D(\frac{5}{2},3)$ (mid-point of side $AB$)
$E(\frac{5}{2},7)$ (mid-point of side $BC$)
$F(4,5)$ (mid-point of side $CA$)

Using the mid-point formula, we can set up the equations for the x-coordinates:
$\frac{x_1+x_2}{2}=\frac{5}{2}\Rightarrow x_1+x_2=5$
$\frac{x_2+x_3}{2}=\frac{5}{2}\Rightarrow x_2+x_3=5$
$\frac{x_3+x_1}{2}=4\Rightarrow x_3+x_1=8$

Subtracting the second equation from the first gives:
$x_1-x_3=0\Rightarrow x_1=x_3$

Substituting this into the third equation:
$2x_1=8\Rightarrow x_1=4$, which also means $x_3=4$.
Then, $x_2=5-4=1$.

Similarly, for the y-coordinates:
$\frac{y_1+y_2}{2}=3\Rightarrow y_1+y_2=6$
$\frac{y_2+y_3}{2}=7\Rightarrow y_2+y_3=14$
$\frac{y_3+y_1}{2}=5\Rightarrow y_3+y_1=10$

Summing these three equations gives:
$2(y_1+y_2+y_3)=30\Rightarrow y_1+y_2+y_3=15$

Solving for each y-coordinate:
$y_3=15-6=9$
$y_1=15-14=1$
$y_2=15-10=5$

Thus, the vertices of the triangle are $A(4,1)$, $B(1,5)$, and $C(4,9)$.

Now, we calculate the lengths of the sides of the triangle $\Delta ABC$:
Side length $a$ (opposite to vertex $A$, length of $BC$):
$a=\sqrt{{(4-1)}^2+{(9-5)}^2}=\sqrt{3^2+4^2}=5$

Side length $b$ (opposite to vertex $B$, length of $AC$):
$b=\sqrt{{(4-4)}^2+{(9-1)}^2}=\sqrt{8^2}=8$

Side length $c$ (opposite to vertex $C$, length of $AB$):
$c=\sqrt{{(1-4)}^2+{(5-1)}^2}=\sqrt{{(-3)}^2+4^2}=5$

The coordinates of the incentre $(h,k)$ are given by the formula:
$h=\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c}$
$k=\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c}$

Substituting the values:
$h=\frac{5\left(4\right)+8\left(1\right)+5\left(4\right)}{5+8+5}=\frac{20+8+20}{18}=\frac{48}{18}=\frac{8}{3}$

$k=\frac{5\left(1\right)+8\left(5\right)+5\left(9\right)}{5+8+5}=\frac{5+40+45}{18}=\frac{90}{18}=5$

Thus, the incentre is $(h,k)=(\frac{8}{3},5)$.

Now, we find the value of $3h+k$:
$3h+k=3\left(\frac{8}{3}\right)+5=8+5=13$.