Given that points P(-6,1), Q(-2, -1) and R(-3,4) find:
a) magnitude of vector RP
b) perimeter of triangle PQR
GIVEN: P(-6 , 1) , Q(-2 , -1) ,
R(-3 , 4).
a. (RP)^2 = (-3 + 6)^2 + (4 - 1) =
9 + 9 = 18,
RP = sqrt(18) = 3*sqrt(2) = 4.24.
b. (PQ)^2 = (-2 + 6)^2 + (-1 , -1)^2 =
= 16 + 4 = 20,
PQ = 2*sqrt(5) = 4.47
(QR)^2 = (-3 + 2)^2 + (4 - 1)^2
= 10,
QR = sqrt(10) = 3.16.
P = PQ + QR + RP = 4.47 + 3.16
+ 4.24 = 11.87 = Perimeter.
NOTE. The distance fohmula is used to cal. the length of each side.
To find the magnitude of vector RP, you can use the distance formula:
a) Magnitude of vector RP = sqrt((x2-x1)^2 + (y2-y1)^2)
Given:
Point P(-6,1) and point R(-3,4)
Substituting the values into the formula, we get:
Magnitude of vector RP = sqrt((-3-(-6))^2 + (4-1)^2)
= sqrt((3)^2 + (3)^2)
= sqrt(9 + 9)
= sqrt(18)
= 3*sqrt(2)
Therefore, the magnitude of vector RP is 3*sqrt(2).
To find the perimeter of triangle PQR, you need to find the lengths of all three sides and then sum them up.
Given:
Point P(-6,1), Q(-2, -1), and R(-3,4)
Using the distance formula to find the length of side PQ:
Length of side PQ = sqrt((x2-x1)^2 + (y2-y1)^2)
= sqrt((-2-(-6))^2 + (-1-1)^2)
= sqrt((4)^2 + (-2)^2)
= sqrt(16 + 4)
= sqrt(20)
= 2*sqrt(5)
Using the distance formula to find the length of side QR:
Length of side QR = sqrt((x2-x1)^2 + (y2-y1)^2)
= sqrt((-3-(-2))^2 + (4-(-1))^2)
= sqrt((-1)^2 + (5)^2)
= sqrt(1 + 25)
= sqrt(26)
Using the distance formula to find the length of side RP:
Length of side RP = sqrt((x2-x1)^2 + (y2-y1)^2)
= sqrt((-3-(-6))^2 + (4-1)^2)
= sqrt((3)^2 + (3)^2)
= sqrt(9 + 9)
= sqrt(18)
= 3*sqrt(2)
Now sum up the lengths of all three sides:
Perimeter of triangle PQR = Length of side PQ + Length of side QR + Length of side RP
= 2*sqrt(5) + sqrt(26) + 3*sqrt(2)
Therefore, the perimeter of triangle PQR is 2*sqrt(5) + sqrt(26) + 3*sqrt(2).
To find the magnitude of vector RP, we need to calculate the length of the line segment connecting point R to point P. The magnitude (or length) of a vector can be found using the distance formula:
The distance formula is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Let's substitute the coordinates of points R and P into the distance formula:
For point R: R(-3, 4)
For point P: P(-6, 1)
Using the distance formula:
d = √[(-3 - (-6))^2 + (4 - 1)^2]
= √[(-3 + 6)^2 + (4 - 1)^2]
= √[3^2 + 3^2]
= √[9 + 9]
= √18
Therefore, the magnitude of vector RP is √18.
To find the perimeter of triangle PQR, we need to calculate the sum of the lengths of all three sides.
The formula to calculate the distance between two points (x1, y1) and (x2, y2) is:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
Using points P, Q, and R:
For side PQ:
d(PQ) = √[(-2 - (-6))^2 + (-1 - 1)^2]
= √[(-2 + 6)^2 + (-1 - 1)^2]
= √[4^2 + (-2)^2]
= √[16 + 4]
= √20
For side QR:
d(QR) = √[(-3 - (-2))^2 + (4 - (-1))^2]
= √[(-3 + 2)^2 + (4 + 1)^2]
= √[(-1)^2 + 5^2]
= √[1 + 25]
= √26
For side RP:
d(RP) = √[(-3 - (-6))^2 + (4 - 1)^2]
= √[(-3 + 6)^2 + (4 - 1)^2]
= √[3^2 + 3^2]
= √[9 + 9]
= √18
Now, we can find the perimeter by adding the lengths of all three sides:
Perimeter = PQ + QR + RP
= √20 + √26 + √18
So, the perimeter of triangle PQR is approximately √20 + √26 + √18.