The line 2x + 3y = 6 meets the y-axis at A and the x-axis at B. C is the point such that AB = BC. CD is drawn perpendicular to AC to meet the line through A parallel to

5x + y = 7 at D.

a. Find the coordinate of A, B and C.

b. State the equations of CD and AD and hence find the coordinates of D.

c. Calculate the area of the triangle ACD.​

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

to find the total number of combinations, multiply the quantity of each item by each other:

2 pairs of shoes x 3 pairs of shorts = 6 different combinations.

1) 2/5

2) zero=0/1

3) zero=0/2

4) -4/5

5) undefined=3/0