ABCD

[IloveManUtd]

ABCD is a square where A(0,2) and C(8,4). AC and BD are diagonals of the square and they intersect at M. Find the coordinates of the points B and D.

How do I find B and D when I only know the gradient? Please help.

 

[mmm4444bot]

Draw a sketch, to visualize the situation.

A square's diagonals intersect at their midpoints. Use the Midpoint Formula, to determine the coordinates of point M.

Calculate the slope of the line through points A and C.

The line through points B and D is perpendicular and passes through point M.

Use the Point-Slope Formula, to find the equation of the line through B and D.

Points B and D must be the same distance from M as points A and C.

Use the equation of the line through B and D, along with the distance formula, to determine the coordinates.

 

[soroban]

Hello, IloveManUtd!

Did you make a sketch?
The answers are quite obvious.

\(\displaystyle ABCD \text{ is a square with }A(0,2)\text{ and }C(8,4).\)
\(\displaystyle AC\) and \(\displaystyle BD\) are diagonals of the square and they intersect at \(\displaystyle M.\)
Find the coordinates of \(\displaystyle B\) and \(\displaystyle D.\)

\(\displaystyle \text{The diagonals of a square bisect each other.}\)
\(\displaystyle M\text{ is the midpoint of }AC\!:\;\;M(4,3)\)

                              C
                              o(8,4)
                  M     *
                  o
     A      *   (4,3) 
 (0,2)o

To go from \(\displaystyle M\) to \(\displaystyle C\), we move: 4 units right and 1 unit up.

                              C
                         (8,4)o
                  M     *     :1
                  o - - - - - +
     A      *   (4,3)   4
 (0,2)o

The diagonals of a square are orthogonal.

To go from \(\displaystyle M\) to \(\displaystyle B\), we move: 4 units up, 1 unit left.
To go from \(\displaystyle M\) to \(\displaystyle D\), we move: 4 units down, 1 unit right.
. . Do you see why?

           B    1
        (3,7)o----+
              *   :
               *  :4          C
                * :           o(8,4)
                 *:     *
                M o
     A     *      :*    4
 (0,2)o            : *
                 4:  *
                  :   *
                  +----o(5,-1)
                    1  D

\(\displaystyle \text{Therefore: }\;B(3,7),\;D(5,-1)\)

 

[IloveManUtd]

Thanks soroban, your explanation is very clear and helpful. Thx.

 

[soroban]

Re: Said yu got ringed

Hello, efgh179!

What planet are you from?