Q1: Find the length and midpoint of segment PQ given P(3,2) and Q(-1,4)
Q2: The midpoint of AB is M(1/2 , 6). Given A(3,7), find the coordinates of B.
for the points (a,b) and (c,d) the midpoint is ((a+c)/2, (b+d)/2)
so midpoint of PQ = ((3-1)/2,(2+4)/2) = (1,3)
for #2
let B be (a,b)
then isn't (a+3)/2 = 1/2
and (b+7)/2 = 6 ?
solve for a and b
A1: To find the length of segment PQ, we can use the distance formula:
Distance Formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of P(3,2) and Q(-1,4), we can plug these values into the formula:
d = √((-1 - 3)^2 + (4 - 2)^2)
= √((-4)^2 + (2)^2)
= √(16 + 4)
= √(20)
= 2√5
So, the length of segment PQ is 2√5.
To find the midpoint of segment PQ, we can use the midpoint formula:
Midpoint Formula:
((x1 + x2)/2 , (y1 + y2)/2)
Using the coordinates of P(3,2) and Q(-1,4), we can plug these values into the formula:
Midpoint = ((3 + (-1))/2 , (2 + 4)/2)
= (2/2, 6/2)
= (1,3)
Therefore, the midpoint of segment PQ is (1,3).
A2: To find the coordinates of point B, we can use the midpoint formula:
Midpoint Formula:
((x1 + x2)/2 , (y1 + y2)/2)
Given the midpoint M(1/2, 6) and point A(3,7), we can plug these values into the formula:
(1/2) = (3 + x)/2 (x-coordinate)
6 = (7 + y)/2 (y-coordinate)
By solving these equations, we can find the coordinates of point B.
Solving for x:
1/2 = (3 + x)/2
1 = 3 + x
x = -2
Solving for y:
6 = (7 + y)/2
12 = 7 + y
y = 5
Therefore, the coordinates of point B are (-2, 5).
To find the length and midpoint of segment PQ given the coordinates P(3,2) and Q(-1,4), we can use the distance formula and the midpoint formula.
1. Length of segment PQ:
The length of a segment can be found using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2)
For our segment PQ, we have P(3,2) and Q(-1,4):
Length of PQ = √((-1 - 3)^2 + (4 - 2)^2)
= √((-4)^2 + (2)^2)
= √(16 + 4)
= √20
= 2√5
Therefore, the length of segment PQ is 2√5.
2. Midpoint of segment PQ:
The midpoint of a segment can be found using the midpoint formula:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
For our segment PQ, we have P(3,2) and Q(-1,4):
Midpoint of PQ = ((3 + (-1)) / 2, (2 + 4) / 2)
= (2 / 2, 6 / 2)
= (1, 3)
Therefore, the midpoint of segment PQ is M(1, 3).
Now, let's move on to the next question.
To find the coordinates of point B given the midpoint M(1/2, 6) and point A(3,7), we can use the midpoint formula.
1. Midpoint formula:
The midpoint of a segment can be found using the midpoint formula:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
For our segment AB, we have M(1/2, 6) and A(3,7):
((3 + x2) / 2, (7 + y2) / 2) = (1/2, 6)
If we solve the equations for x2 and y2, we can find the coordinates of B.
(3 + x2) / 2 = 1/2
3 + x2 = 1
x2 = 1 - 3
x2 = -2
(7 + y2) / 2 = 6
7 + y2 = 12
y2 = 12 - 7
y2 = 5
Therefore, the coordinates of point B are (-2, 5).
I hope this helps! Let me know if you have any further questions.