SOLUTION: P(-1,-2), Q(5,k), R(8,2), S(h,1) are the four vertices of the parallelogram PQRS, Find the value of h, and the value of k.

Algebra ->  Parallelograms -> SOLUTION: P(-1,-2), Q(5,k), R(8,2), S(h,1) are the four vertices of the parallelogram PQRS, Find the value of h, and the value of k.      Log On


   

Question 945242: P(-1,-2), Q(5,k), R(8,2), S(h,1) are the four vertices of the parallelogram PQRS, Find the value of h, and the value of k.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given:
P(-1,-2), Q(5,k), R(8,2), S(h,1) are the four vertices of the parallelogram PQRS,
the parallelogram is quadrilateral with two pairs of parallel and equal in length sides
PQ is parallel and equal to RS
QR is parallel and equal to PS
distance from P and Q is equal to distance from S to+R
and
distance from P and S is equal to distance from Q to R
Now use the distance formula to find the distance between the two points, P(-1,-2) and Q(5,k):

d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%29%5E2%29
d=sqrt%28%28-1-5%29%5E2%2B%28-2-k%29%29%5E2%29
d=sqrt%2836%2Bk%5E2%2B4k%2B4%29
d=sqrt%28k%5E2%2B4k%2B40%29 => this is also equal to distance from R(8,2) to S(h,1)
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%29%5E2%29
d=sqrt%28%288-h%29%5E2%2B%282-1%29%29%5E2%29
d=sqrt%28h%5E2-16h%2B64%2B1%29
d=sqrt%28h%5E2-16h%2B65%29
since PQ is parallel and equal to RS, we have
sqrt%28k%5E2%2B4k%2B40%29=sqrt%28h%5E2-16h%2B65%29 ......square both sides
%28sqrt%28k%5E2%2B4k%2B40%29%29%5E2=%28sqrt%28h%5E2-16h%2B65%29%29%5E2
k%5E2%2B4k%2B40=h%5E2-16h%2B65 ........eq.1

now find the distance between the two other points Q(5,k) and R(8,2)
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%29%5E2%29
d=sqrt%28%285-8%29%5E2%2B%28k-2%29%29%5E2%29
d=sqrt%289%2Bk%5E2-4k%2B4%29
d=sqrt%28k%5E2-4k%2B13%29
then, between the two other points P(-1,-2) and S(h,1)
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%29%5E2%29
d=sqrt%28%28-1-h%29%5E2%2B%28-2-1%29%29%5E2%29
d=sqrt%28h%5E2%2B2h%2B1%2B9%29
d=sqrt%28h%5E2%2B2h%2B10%29
so,
sqrt%28k%5E2-4k%2B13%29=sqrt%28h%5E2%2B2h%2B10%29
k%5E2-4k%2B13=h%5E2%2B2h%2B10........eq.2
solve the system:
k%5E2%2B4k%2B40=h%5E2-16h%2B65 ........eq.1
k%5E2-4k%2B13=h%5E2%2B2h%2B10........eq.2
-------------------------------------------subtract 2 from 1
k%5E2%2B4k%2B40-%28k%5E2-4k%2B13%29=h%5E2-16h%2B65-%28h%5E2%2B2h%2B10%29
k%5E2%2B4k%2B40-k%5E2%2B4k-13=h%5E2-16h%2B65-h%5E2-2h-10
8k%2B27=-18h%2B55 ...solve for k
8k=-18h%2B55-27
k=-18h%2F8%2B28%2F8
k=-2.25h%2B3.5 .............substitute in eq.1

%28-2.25h%2B3.5%29%5E2%2B4%28-2.25h%2B3.5%29%2B40=h%5E2-16h%2B65 ........eq.1 ...solve for h
5.0625h%5E2-15.75h%2B12.25-9h%2B14%2B40=h%5E2-16h%2B65
4.0625h%5E2-8.75h%2B1.25+=+0+.......use quadratic formula

h+=+%288.75+%2B-+sqrt%28+76.5625-20.3125+%29%29%2F8.125+
h+=+%288.75+%2B-+sqrt%28+56.25+%29%29%2F8.125+

h+=+%288.75+%2B-+7.5%29%2F8.125+

h+=+2 or h+=+0.153846
we will use highlight%28h+=+2%29 and find k
k=-2.25%2A2%2B3.5+
k=-4.5%2B3.5

highlight%28k=-1%29
so, we will have points:
P(-1,-2), Q(5,-1), R(8,2), S(2,1)
now we can graph a parallelogram PQRS