Explanation:
let me write the information in clear text :
P = (A, -11)
Q = (5, B)
R = (2, 5)
S = (1, 1)
this is a parallelogram. that means every pair of opposite sides must be parallel. and that means they must have the same slope.
the slope of a line is the ratio
y coordinate difference / x coordinate difference
when going from one point on the line to another.
the slope of RS is
(5 - 1)/(2-1) = 4/1 = 4
the slope of PQ must be the same.
(-11 - B)/(A - 5) = 4
-11 - B = 4(A - 5) = 4A - 20
-B = 4A - 9
B = -4A + 9
and then 2 other 2 sides must be parallel (have the same slope) :
slope PR = slope QS
or
slope PS = slope QR
let's face it, there are 2 solutions for that reason, as we don't know which vertices are connected to which.
slope PR and QS :
(-11 - 5)/(A - 2) = (B - 1)/(5 - 1)
-16×4 = (B - 1)(A - 2) = AB - 2B - A + 2
-64 = AB - 2B - A + 2
0 = AB - 2B - A + 66
now, let's use our first identity from the slope of PQ in here :
0 = A(-4A + 9) - 2(-4A + 9) - A + 66
0 = -4A² + 9A + 8A - 18 - A + 66 = -4A² + 16A + 48
0 = -A² + 4A + 12 = A² - 4A - 12
remember the general solution of a quadratic equation
ax² + bx + c = 0
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = -4
c = -12
x = A
A = (4 ± sqrt((-4)² - 4×1×-12))/(2×1) =
= (4 ± sqrt(16 + 48))/2 = (4 ± sqrt(64))/2 =
= (4 ± 8)/2 = 2 ± 4
A1 = 2 + 4 = 6
A2 = 2 - 4 = -2
out of
B = -4A + 9
we get
B1 = -4×6 + 9 = -24 + 9 = -15
B2 = -4×-2 + 9 = 8 + 9 = 17
now for slope PS and QR
(-11 - 1)/(A - 1) = (B - 5)/(5 - 2)
-12×3 = (B - 5)(A - 1) = AB - B - 5A + 5
-36 = AB - B - 5A + 5
0 = AB - B - 5A + 41
now using again the identity of the PQ slope ;
0 = A(-4A + 9) - (-4A + 9) - 5A + 41
0 = -4A² + 9A + 4A - 9 - 5A + 41 = -4A² + 8A + 32
0 = -A² + 2A + 8 = A² - 2A - 8
again solving the quadratic equation :
a = 1
b = -2
c = -8
x = A
A = (2 ± sqrt((-2)² - 4×1×-8))/(2×1) =
= (2 ± sqrt(4 + 32))/2 = (2 ± sqrt(36))/2 =
= (2 ± 6)/2 = 1 ± 3
A1 = 1 + 3 = 4
A2 = 1 - 3 = -2
and from
B = -4A + 9
we get
B1 = -4×4 + 9 = -16 + 9 = -7
B2 = -4×-2 + 9 = 8 + 9 = 17
so we get the possible solutions
1. P(6, -11), Q(5, -15)
2. P(-2, -11), Q(5, 17)
3. P(4, -11), Q(5, -7)
4. P(-2, -11), Q(5, 17)
the second criteria for a parallelogram is that the opposite sides must be equally long.
RS = sqrt((5 - 1)² + (2 - 1)²) = sqrt(16 + 1) = sqrt(17)
from the possible solutions we found solutions 2 and 4 are identical and their PQ side length would be
PQ = sqrt((17 - -11)² + (5 - -2)²) = sqrt(28² + 7²) = sqrt(833)
which is NOT sqrt(17).
so, these solutions are out.
but solutions 1 and 3 have the correct side length :
PQ = sqrt((-15 - -11)² + (5 - 6)²) = sqrt((-4)² + (-1)²) = sqrt(17)
and
PQ = sqrt((-7 - -11)² + (5 - 4)²) = sqrt(4² + 1²) = sqrt(17)
so, the correct values of A and B are
A = 6, B = -15
A = 4, B = -7