Vtyjkyuk

Can you help me with this, I’m struggling with finding the right equation for this problem.

Specifically, I need to find the co-ordinates of D.

There is not a direct formula, or at least this is irrelevant if you use the ideas in the image.

The slope of the line $BC$ is:
$$a=fracy_2-y_1x_2-x_1=frac4-(-5)-5-1=-frac32.$$
Since $AD||BC$, the slope of the line $AD$ is also $-frac32$. The equation of the line $AD$ must pass through the point $A(3,5)$:
$$y_AD=-frac32x+frac192.$$
Since $ADC=90^circ$, it implies $CDperp AD$, hence the line $CD$ has the slope $frac23$ and it passes through the point $C(1,-5)$:
$$y_CD=frac23x-frac173.$$
Finally, the intersection of the lines $AD$ and $CD$ is the point $D$:
$$begincasesy_AD=-frac32x+frac192\ y_CD=frac23x-frac173endcases Rightarrow D(x,y)=D(7,-1).$$

Let $(x, y)$ be the coordinates of point $D$.

Since $overlineAD parallel overlineBC$, the slope of $overlineAD$ is equal to that of $overlineBC$. Since,
beginalign*
m_overlineAD & = fracy - 5x - 3\
m_overlineBC & = frac-5 - 41 - (-5) = frac-96 = -frac32
endalign*
we obtain
$$fracy - 5x - 3 = -frac32 tag1$$
Since $mangle ADC = 90^circ$, $overlineAD perp overlineCD$. Since
$$m_overlineAD = m_overlineBC = -frac32$$
and the slopes of nonvertical perpendicular lines are negative reciprocals,
$$m_overlineCD = frac23$$
Since the slope of $overlineCD$ is
$$m_overlineCD = fracy - (-5)x - 1 = fracy + 5 x - 1$$
Equating the two results for the slope of $overlineCD$ yields
$$fracy + 5x - 1 = frac23 tag2$$
You can determine the coordinates of point $D$ by solving the system of equations 1 and 2.