Coordinate geometry help?

Question

Consider a quadrilateral ABCD in which A is (0,10). B is (2,16) and C is (8,14).
(i) Show that triangle ABC is isosceles
The point D lies on the x-axis and is such that AD=CD
Find
(ii) Coordinates of D
(iii) Ratio of area of triangle ABC to the area of triangle ACD

Answer 1

i) you can show that 3 sides are equal.
Do this for 3 times:
AB=B-A=(2,6)
length of AB=sqrt(4+36)=2sqrt(10)

the length of BC and CA should also be 2sqrt(10)

ii)AD=AC
the coordinate of D will be (0,x)
use two length equation and equal them....
then you can get x

iii) simple division

Answer 2

Consider a quadrilateral ABCD in which A is (0,10). B is (2,16) and C is (8,14).

(i) Show that triangle ABC is isosceles.

AB = √[(2-0)² + (16-10)²] = √(4 + 36) = √40
BC = √[(8-2)² + (14-16)²] = √(36 + 4) = √40

Two sides are of equal length. Therefore the triangle is congruent.
______________

The point D lies on the x-axis and is such that AD=CD

(ii) Find the coordinates of D.
D(x,0)

AD = √[(x-0)² + (0-10)²] = √(x² + 100)
CD = √[(x-8)² + (0-14)² = √(x² + 16x + 260)

x² + 100 = x² - 16x + 260
16x = 160
x = 10

D(10,0)
_______________

iii) Find the ratio of area of triangle ABC to the area of triangle ACD.

Use the cross product.

AB = <2, 6, 0>
AC = <8, 4, 0>
AD = <10, -10, 0>

Area ΔABC = (1/2) || AB X AC ||
= (1/2) || <2, 6, 0> X <8, 4, 0> || = (1/2) || -40k || = 20

Area ΔACD = (1/2) || AC X AD ||
= (1/2) || <8, 4, 0> X <10, -10, 0> || = (1/2) || -120k || = 60

Ratio (Area ΔABC) / (Area ΔACD) = 20/60 = 1/3
________