ABCD is a rectangle and A and B are the points (4,2) and (2,8) respectively. Given that the equation of AC is y=x-2, find a) Equation of BC b) coordinates of B and C

A(4,2), B(2,8)

AC: y=x-2

Point point form for a line through (a,b) and (c,d) is (c-a)(y-b)=(d-b)(x-a)

AB is  (2 - 4)(y - 2) = (8-2)(x-4)

-2(y - 2)=6(x-4)

y - 2 = -3(x-4)

y = -3x + 14

BC is perpendicular through B, so slope 1/3 and we calculate the constant as y-(1/3)x:

y = (1/3) x  + (8 - (1/3)(2) ) = (1/3) x + 22/3

a)  BC is  y =  (1/3) x + 22/3

C is the meet of AC and BC,

x - 2 = (1/3) x + 22/3

3x - 6 = x + 22

2x = 28

x = 14

y = x-2 = 12

Check: y = (1/3)x + 22/3 = (14+22)/3 = 36/3 = 12  good

C(14,12)

The remaining corner is D.   We have A-B=D-C or

D = A+C-B+C = (4,2)+(14,12)-(2,8) = (16, 6)

b) B(2,8) [given, but asked for]  C(14,12), D(16,6)

56.6

step-by-step explanation:

the ratio of the perimeters is equal to the scale.

the ratio of the areas is equal to the square of the scale.

34 / 20 = k

a / 19.6 = k²

substituting:

a / 19.6 = (34 / 20)²

a = 19.6 (34 / 20)²

a ≈ 56.6

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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