Show that PQRS is a rectangle.
(P: (1,4); Q: (2,6); R: (8,3); S: (7,1)
sorry about my arithmetic error
when I copied your diagram to paper, I had point
Q as (3,6) instead of (2,6)
I did this for you , it was #5 of
http://www.jiskha.com/display.cgi?id=1386029033
Ok, once more, very simple:
slope PQ = (6-4/(2-1) = 2
slope QR = (6-3)/(2-8) = -3/5
So there is no right angle at Q
but a rectangle must have 4 right-angles
I have shown that one of them is not, so there is no point to continue
It is NOT a rectangle
P: (1,4); Q: (2,6); R: (8,3); S: (7,1)
PQ = (6-4)/(2-1) = 2/1 = 2
QR = (3-6)/(8-2) = -3/6 = -1/2
RS = (1-3)/(7-8) = 2
PS = (1-4)/(7-1) = -3/6 = -1/2
The slopes of all of consecutive sides are negative reciprocals. Therefore, quadrilateral PQRS with 4 angles is a rectangle.
To show that PQRS is a rectangle, we need to demonstrate that all four sides of the quadrilateral are equal in length and that the opposite sides are parallel.
First, let's find the lengths of all four sides:
Side PQ: Using the distance formula √((x2 - x1)^2 + (y2 - y1)^2), we can calculate the distance between points P(1,4) and Q(2,6):
d(PQ) = √((2 - 1)^2 + (6 - 4)^2)
= √(1^2 + 2^2)
= √(1 + 4)
= √5
Side QR: Using the distance formula, we can calculate the distance between points Q(2,6) and R(8,3):
d(QR) = √((8 - 2)^2 + (3 - 6)^2)
= √(6^2 + (-3)^2)
= √(36 + 9)
= √45
= 3√5
Side RS: Using the distance formula, we can calculate the distance between points R(8,3) and S(7,1):
d(RS) = √((7 - 8)^2 + (1 - 3)^2)
= √((-1)^2 + (-2)^2)
= √(1 + 4)
= √5
Side SP: Using the distance formula, we can calculate the distance between points S(7,1) and P(1,4):
d(SP) = √((1 - 7)^2 + (4 - 1)^2)
= √((-6)^2 + 3^2)
= √(36 + 9)
= √45
= 3√5
Since the lengths of all four sides are equal (PQ = QR = RS = SP = √5), we have shown that the quadrilateral PQRS has equal sides.
Next, let's examine the slopes of opposite sides to determine if they are parallel.
The slope of side PQ:
m(PQ) = (y2 - y1) / (x2 - x1)
= (6 - 4) / (2 - 1)
= 2 / 1
= 2
The slope of side RS:
m(RS) = (y2 - y1) / (x2 - x1)
= (1 - 3) / (7 - 8)
= (-2) / (-1)
= 2
Since the slopes of side PQ and RS are equal (m(PQ) = m(RS) = 2), we have shown that the opposite sides are parallel.
Hence, based on the verification of equal side lengths and parallel opposite sides, we can conclude that PQRS is a rectangle.