Answer:
x=-7
y=11
Step-by-step explanation:
Perpendicular Vectors
Two vectors defined as their endpoints [tex]\vec u=<a,b>[/tex] and [tex]\vec v=<c,d>[/tex] are perpendicular if their dot product a.b is zero. The dot product is
[tex]\vec u.\vec v=ac+bd[/tex]
In other words
ac+bd=0
Let's treat all the points as the extremes of vectors, so we can easily find the missing coordinates
B (3,5) and C (7,15) define a segment, the vector
[tex]\overrightarrow{BC}=<7-3,15-5>=<4,10>[/tex]
The point A is A (x,9), we need to form a vector with B
[tex]\overrightarrow{AB}=<x-3,9-5>=<x-3,4>[/tex]
this vector must be perpendicular to BC, so, applying the dot product we have
[tex]4(x-3)+40=0[/tex]
[tex]4x-12=-40[/tex]
[tex]x=-7[/tex]
The point D is D(17,y), we need to form a vector with C
[tex]\overrightarrow{CD}=<17-7,y-15>=<10,y-15>[/tex]
this vector must be perpendicular to BC, so, applying the dot product we have
[tex](4)(10)+(10)(y-15)=0[/tex]
[tex]40+10y-150=0[/tex]
[tex]10y=110[/tex]
[tex]y=11[/tex]
The points are
[tex]\boxed{A(-7,9), D(17,11)}[/tex]
