1. ## Need Help with this quadratics stuff

Diagram attatched
Find:
i) the equation of the line through points A and D

ii) the coordinates of D

iii) the equation of the perpendicular bisector of AB

iv) the coordinates of C

v) the area of triangle ADC

vi) the area of the quadrilateral ABCD

P.S Angle BAD is a right angle and C lies on the perpendicular bisector of AB. The equation of the line through points B and C is 3y=4x-14
You dont have to answer all question any would be nice. Thanks

2. ## Coordinate Geometry

Hello scubasteve94
Originally Posted by scubasteve94
Diagram attatched
Find:
i) the equation of the line through points A and D

ii) the coordinates of D

iii) the equation of the perpendicular bisector of AB

iv) the coordinates of C

v) the area of triangle ADC

vi) the area of the quadrilateral ABCD

P.S Angle BAD is a right angle and C lies on the perpendicular bisector of AB. The equation of the line through points B and C is 3y=4x-14
You dont have to answer all question any would be nice. Thanks
Not quite sure why you've called it Quadratics, but here goes:

(i) Gradient of AB = $\displaystyle \frac{6-8}{8-2} = -\tfrac{1}{6}$

$\displaystyle \Rightarrow$ gradient of AD = 6

$\displaystyle \Rightarrow$ equation of AD is $\displaystyle y-8=6(x-2)$

i.e. $\displaystyle y = 6x-4$

(ii) D is intercept of this line with y-axis; i.e. (0, -4)

(iii) Perpendicular bisector of AB also has gradient 6, and passes through the mid-point of AB, i.e. (5, 7). So its equation is

$\displaystyle y -7=6(x-5)$

i.e. $\displaystyle y = 6x -23$

(iv) The line in (iii) meets AC at C, where the equations are simultaneously true; i.e. where

$\displaystyle 3(6x-23)=4x-14$

$\displaystyle \Rightarrow 18x - 69 = 4x-14$

$\displaystyle \Rightarrow 14x = 55$

$\displaystyle \Rightarrow x=\frac{55}{14}$

I have to go now. Perhaps someone else can pick this up?

3. Originally Posted by scubasteve94
i need to find the equation of the line through points A and D so..
...

1. Which kind of quadrilateral do you have. (According to your sketch it could be a rectangle(?))

2. Is the point D placed on the y-axis?

4. It does not specify what kind of quadrilateral it is but i dont think it is a rectangle.

Yes D is on the y axis

5. Originally Posted by scubasteve94
It does not specify what kind of quadrilateral it is but i dont think it is a rectangle.

Yes D is on the y axis
The slope of the line passing through A and B is:

$\displaystyle m_{AB}=\dfrac{6-8}{8-2}=-\dfrac13$

Since $\displaystyle AB \perp AD$ the slope of the line AD is:

$\displaystyle m_{AD}=-\dfrac1{m_{AB}}=3$

Now use the point-slope-formula of a straight line to determine the equation of the line AD:

$\displaystyle y-8=3(x-2)~\implies~\boxed{y=3x+2}$

And now you know that D(0, 2)

6. I now have the equation of the perpendicular bisector of AB but i am unsure how to use this to find the coordinates of C. Grandad subbed it into the equation of BC whis is 3y=4x-14 but i am unsure why he did this?

[SIZE=3]Hello scubasteve94Not quite sure why you've called it Quadratics, but here goes:

(i) Gradient of AB = $\displaystyle \frac{6-8}{8-2} =\bold{\color{red} -\tfrac{1}{6}}$

...
I don't want to pick at you but ...

8. Originally Posted by scubasteve94
It does not specify what kind of quadrilateral it is but i dont think it is a rectangle.

Yes D is on the y axis
If the quadrilateral is actually a rectangle then the midpoint of BD must be the midpoint of AC too:

$\displaystyle M_{BD}\left(\frac{6+0}2\ ,\ \frac{8+2}2\right)$ that means:$\displaystyle M_{BD}\left(3, 5\right)$

$\displaystyle M_{BD} = M_{AC}\left(\frac{x_C+2}2\ ,\ \frac{y_C+8}2\right)$

Therefore $\displaystyle x_C=4, y_C=2$ that means the point C is $\displaystyle C(4, 2)$

9. i no i also picked up on that and have made the correction to my notes!!
but im still unsure how to find the coordinates of C

10. Originally Posted by earboth
If the quadrilateral is actually a rectangle then the midpoint of BD must be the midpoint of AC too:
yes that is true but it does not specify that it is a rectangle so..

11. Originally Posted by scubasteve94
i no i also picked up on that and have made the correction to my notes!!
but im still unsure how to find the coordinates of C
1. Calculate the equation of the perpendicular bisector of AB:
$\displaystyle M_{AB}(5, 7)$ Slog of the perpendicular bisector is m = 3 (compare my previous post)

Then $\displaystyle b: y - 7 = 3(x-5)~\implies~\boxed{y=3x-8}$

C is situated on the bisector and the line $\displaystyle BC: 3y=4x-14$

Calculate the intersection point:

$\displaystyle 3(3x-8)=4x-14~\implies~5x=10~\implies~x=2$ . Plug in this value into the first equation: $\displaystyle y = 3\cdot 2 - 8 = -2$

Thus C(2, -2)

12. ## Coordinate Geometry

Hello again -

Sorry for my silly error earlier. Of course 6 - 8 = -2, not -1.

(While I was teaching, I always used to tell my students that 99% of computational errors were either due to a minus sign or a factor of 2. So it's a factor of 2, then, this time!)

Anyway, let's complete this answer now.

Of course, as earboth has shown, D is (0, 2) and C is (2, -2). So the diagram is like the one I have attached. So:

(v) The area of triangle ADC $\displaystyle = \tfrac{1}{2}AC\cdot DE = \tfrac{1}{2}10\cdot 2 = 10$ sq units.

(vi) The area of triangle ABC = $\displaystyle \tfrac{1}{2}AC\cdot BF = \tfrac{1}{2}10\cdot 6 = 30$ sq units. So area of quadrilateral ABCD = 40 sq units.