SOLUTION: 1. Kite ABCD has 2 of its vertices at A(-3,8) and B(7,14). 1(a). Given that the equation of the longer diagonal BD is y=4x-14, find the equation of the short diagonal AC express

We just help you with the first problem. Post again for the others.

 

1(a). Given that the equation of the longer diagonal BD is y=4x-14, find the
equation of the short diagonal AC expressing your answer in the form ax+by+c=0.
 
To write the equation of AC we must have its slope and a point it goes through.
We already have a point it goes through, namely A(-3,8). Now all we need is it
slope.  We know that AC is perpendicular to BD, and we know that to find the
slope of a second line perpendicular to a first line, we invert the slope of
the first line and change its sign.  

The long diagonal BD has equation y = 4x-14.  We compare that to y=mx+b
and see that BD has slope m=4.  Since the short diagonal AC is perpendicular
to the long diagonal BD, then AC has slope .  Next, to find its
equation we use the point-slope form: 

    y - y1 = m(x - x1)

      y - 8 = [x - (-3)]

      y - 8 = (x +3)

Multiply through by 4:

    4y - 32 = -1(x + 3)

    4y - 32 = -x - 3

x + 4y - 29 = 0

That's the equation of AC in the form ax+by+c=0

1(b). Find the co-ordinates of E, which is the point of intersection.

To do that we solve the system of equations:

          y = 4x - 14
x + 4y - 29 = 0

by substitution and get (x,y) = (5,6), so we can label point E as E(5,6)

 





 
1(c). Hence establich the co-ordinates of C. 

To establish the coordinates of point C we realize that E(5,6) is

8 units to the right of point A(-3,8), since 5-(-3) = +8, and also
that E is 2 units below point A((-3,8) since 6-9 = -2.

So therefore point C must be 8 units right and 2 units below point
E(5,6).  Since 5+8=13, and 6-2 = 4, then point C must be (13,4), so
we can fill in the coordinates of point C:

 

The kite is the red figure below, although we are not given enough information
to find the coordinates of D, so we can only leave that point as just D(?,?). 

 

Edwin