Find the coordinates????

Question

Quadrilateral ABCD:

 

the point A is (−1, −1),

the point B is (3, 6) and

the point C is (9, 4).

D is unknown but below the x axis.

 

 

The diagonals AC and BD intersect at M.

Angle BMA = 90° and BM = MD.

 

(sorry, there's no diagram)

 

Find the coordinates of D.

Answer 1

Since the question intentionally did not provide a diagram, you need to plot the given points on a coordinate system and connect the points so that you get a closed polygon resembling a quadrilateral.  

 

First step:

 

Draw the three coordinates which is A, B, and C.  Then connect the points in the order given.  Be sure to label your points.  You should end up with 2 lines segments.

 

 

Second step:

 

Use the angle and diagonals as your references to locate point M.   AC and BD are line segments that intersect at point M.  So you are going to connect point A and point C to create line segment AC.  Then create a line segment from point B that is perpendicular to line segment AC.  You will then have two triangles.  Notice that perpendicular lines form right angles.  Right angles are always 90 degrees.  That 90 degree angle that you created is angle M.  Triangle BMA is the right triangle.  That right triangle contains point M.  Now label that point.

 

Third step:

 

Draw a line segment from point M so that it is the same distance as BM.  The endpoint of this segment that you draw is point D and it should be below the x-axis.  Since AC and BD intersects at point M and BM=MD, point M is the midpoint of BD.

 

Fourth Step:

 

Use the midpoint formula to find point D.

 

(x_M , y_M) = ( (x_B + x_D) / 2 , (y_B + y_D) / 2 )

 

 

where:

midpoint = point M = (x_M , y_M)

Point B = (x_B , y_B)

Point D = (x_D , y_D)

 

 

Break up this formula into two equations.  One equation to find the x-coordinate of D, and the other to find the y coordinate of D.

 

x_M = (x_B + x_D) / 2                and                   y_M =  (y_B + y_D) / 2

 

 

x_D = 2(x_M) - x_B                    and                   y_D = 2(y_M) - y_B

 

 

Just follow the steps and you will get you point D.  If you any questions, let me know

Answer 2

Hi Chittar.

 

So you have diagonals AC and BD. You have points A, B, and C defined. You don't know point D yet.

 

A(-1,-1)

B(3,6)

C(9,4)

D(x,y)

 

You know a few things:

 

1) You know points A and C make a line. You also know both points A and C, so you can determine the equation of the line between them.

 

2) You know that AC and BD intersect at a 90 degree angle. That means the diagonals are perpendicular. That tells you the slope of AC is the negative reciprocal of the slope of BD.

 

So.

 

For line AC:

 

m = (y2 - y1)/(x2 - x1) = 5/10 = 1/2

 

y - y1 = m(x - x1)

 

y - -1 = 1/2(x - -1)

 

y + 1 = 1/2(x + 1)

 

y = 1/2x - 1/2

 

For line BD:

 

m = the negative reciprocal of 1/2, which is -2

 

y - y1 = m(x - x1)

 

y - 6 = -2(x - 3)

 

y - 6 = -2x + 6

 

y = -2x + 12

 

Cool. Now we have two lines. When they intersect, the y's will be equal to each other.

 

y = 1/2x - 1/2

y = -2x + 12

 

1/2x - 1/2 = -2x + 12

 

5/2x = 25/2

 

x = 5

 

Now plug this into one of your equations to find y.

 

y(5) = -2(5) + 12

 

= -10 + 12

 

= 2

 

So point M has an x value of 5 and a y value of 2

 

M(5, 2)

 

So now we want to find the distance from B to M.

 

d_BM = √((2 - 6)² + (5 - 3)²)

 

= √((-4)² + (2)²)

 

= √(20)

 

Cool. We have a distance BM. Your problem also tells us that the distance MD = BM. So now we know that:

 

d_MD = √(20) = √((y-2)² + (x-5)²)

 

Square both sides.

 

20 = (y-2)² + (x-5)²

 

This looks like it'll have infinite solutions won't it? Fortunately we know something else from earlier. The line that intersects through B and D has the equation:

 

y = -2x + 12

 

So substitute for y.

 

20 = (-2x + 12 - 2)² + (x-5)²

 

70 = (-2x+10)² + (x-5)²

 

70 = 4x² - 40x + 100 + x² - 10x + 25

 

5x² - 50x + 55 = 0

 

5(x² - 10x + 11) = 0

 

5(x-11)(x+1) = 0

 

x = 11,-1

 

Your point D is below the x-axis, meaning the y value will be negative. Which of these two x values makes the y value negative?

 

y(11) = -2(11) + 12 = -10

 

Yep.

 

y(-1) = -2(-1) + 12 = 14

 

Nope.

 

Your point D is (11,-10)

 

Hope this helps.

Answer 3

First we will find slope of line AC

the point A is (−1, −1),

the point C is (9, 4).

 

So slope of line AC = m1=4-(-1)/9-(-1)=5/10=1/2

 

Equation of line AC is

y-4=1/2(x-9)

2y-8=x-9

2y=x-1__________eq 1

 

The diagonals AC and BD intersect at M.
Angle BMA = 90°

it says BD and AC are perpendicular

So  slope of BD is =-2

And line passing through point B(3,6)

so equation of BD is

y-6=-2(x-3)

y-6=-2x+6

y=-2x+12 _________eq 2

 

We can get co-ordinate of intersection point M by solving eq 1 and eq2

 

substitute y=-2x+12 in eq 1

 

2y=x-1

 

2(-2x+12)=x-1

-4x+24 =x-1

25=5x

5=x

 

substitute x=5 in eq 2

y=-2x+12

y=-2(5)+!2=-10+12=2

 

so coordinates of m (5,2)

 

given that

BM = MD

M is midpoint of BD

 

so using midpoint formula

 

(5,2) =[(3+x)/2,(6+y)/2]

 

(3+x)/2=5      

3+x=10          

x=7

 

(6+y)/2=2

6+y=4

y=-2

 

so co-ordinates of D(7,-2)