help! trig homework!?

Question

1. find the area of the rectangle with the vertices A(-5,1), B(-3,-1), C(3,5) and D(0,3)
2. A quadrilateral is a rhombus if all four sides are equal in length. If P(1, 2), Q(4, -2), R(1,-6), and S are consecutive vertices of a rhombus, what are the coordinates of Point S

Answer 1

1. Q(A_1, A_2) = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2), formula for length of a vertical then
ab=4
bc=sqrt(36+36)=6*sqrt(2)
area=ab*bc=24sqrt(2)
2. pq=sqr(9+16)=5
rq=sqr(9+16)=5
rs=(1-xs)^2+(-6-xy)^2=5
ps=(1-xs)^2+(2-xy)^2=5
solve the last 2 equations for xs and ys

Answer 2

These are not the problems of trigonometry but of co-ordinate geometry.

( 1 ) Find lengths AB, BC, CD and DA by distance formula. You get AB = 2 root 2, BC = 6 root 2, CD = root 13, DA = root 29. This does not form a rectangle. Problem is erroneous.

( 2 ) By distance formula, PQ = 5, QR = 5
Let s = ( x, y )

Now, PR and QS are diagonals of rhombus which should bisect each other, i.e., their midpoints are equal.

So, x + 4 = 1 + 1 and y - 2 = 2 - 6

which gives x = -2 and y = -2

So S = ( -2, -2 )

Check RS = SP = 5. So answer S = ( -2, -2 ) is correct.

Answer 3

Question 1
AB² = (- 3 + 5)² + (- 1 -1)²
AB² = 4 + 4
AB = √8

BC² = (3 + 3)² + (5 + 1)²
BC² = 36 + 36
BC = √72

Area = √8 x √72
Area = 2.√2 x 6.√2
Area = 12 x 2
Area = 24 units²

Question 2
S is point S(x,y)
Vector QR = (- 3 , - 4)
Vector PS = (x - 1 , y - 2)
x - 1 = - 3
x = - 2

y - 2 = - 4
y = - 2

S is point S(-2 , -2)

Answer 4

1.) you can use determinants in finding the area of the vertices.
2.) if you'll plot those points on the cartesian plane, i think you can solve the problem...