points P(4,12), Q(9,14), and R(13,4)are the three vertices of a rectangle.

a. find the coordinates of the fourth vertex, S.
b. write a short description of how you found the coordinates of S.
c. find the lengths of the diagonals of rectangle PQRS. COMMENT on your results.
pls help me

a. To find the coordinates of the fourth vertex, S, we can use the properties of rectangles. In a rectangle, opposite sides are equal in length and parallel, and the diagonals are equal in length and bisect each other.

First, let's find the length of side PQ using the distance formula:

Distance PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
P(4,12) and Q(9,14)
PQ = √[(9 - 4)² + (14 - 12)²]
PQ = √[25 + 4]
PQ = √29

Since the opposite sides of a rectangle are equal in length, the length of SR will also be √29.

Next, let's find the length of side QR using the distance formula:

Distance QR = √[(x₂ - x₁)² + (y₂ - y₁)²]
Q(9,14) and R(13,4)
QR = √[(13 - 9)² + (4 - 14)²]
QR = √[16 + 100]
QR = √116
QR = 2√29

Since the opposite sides of a rectangle are equal in length, the length of PS will also be 2√29.

b. To find the coordinates of the fourth vertex, S, we know that the length of PS is equal to QR, and the length of PQ is equal to SR. So, we can take the coordinates of Q(9,14) and add/subtract the distances we found:

To find S(?, ?), we add -√29 to Q(x,y):
S(x, y) = (9 - √29, 14 - 2√29)
Therefore, the coordinates of the fourth vertex, S, are (9 - √29, 14 - 2√29).

c. The lengths of the diagonals of rectangle PQRS can be found using the distance formula:

To find the length of diagonal PR:
Distance PR = √[(x₂ - x₁)² + (y₂ - y₁)²]
P(4,12) and R(13,4)
PR = √[(13 - 4)² + (4 - 12)²]
PR = √[81 + 64]
PR = √145

To find the length of diagonal QS:
Distance QS = √[(x₂ - x₁)² + (y₂ - y₁)²]
Q(9,14) and S(9 - √29, 14 - 2√29)
QS = √[(9 - (9 - √29))² + (14 - (14 - 2√29))²]
QS = √[√29² + 2√29²]
QS = √[29 + 58]
QS = √87

Comment: The lengths of the diagonals, PR and QS, are √145 and √87 respectively. Since the diagonals of a rectangle are equal in length, these values should be the same. However, in this case, √145 ≠ √87. This means that the given points do not form a rectangle. There might be an error in the given coordinates or the assumption that the points form a rectangle. Please double-check the given points or provide additional information if possible.