Use the diagram below for the following question.
P(2,1), Q(11,1), and R(8,7)
M and N are the midpoints ofandrespectively.
Find MN and PQ.
How do these lengths compare with each other?.
Which points are M and N the midpoints of?
"M and N are the midpoints ofandrespectively"
From the context of the problem I will assume you meant
"M and N are the midpoints of RP and RQ respectively"
M is (5,4) and N is (9.5 , 4)
MN = √((9.5-5)^2 + (4-4)^2) = 4.5
PQ = √((11-2)^2 + 0^2) = 9
So what do you think?
( I suggest you also check the slopes of MN and PQ )
To find the length of MN and PQ, we can use the distance formula. The distance formula calculates the length of a line segment between two points.
The distance formula is given by:
√[(x2 - x1)^2 + (y2 - y1)^2]
Let's find the length of MN first. We can use P(2,1) and Q(11,1) to find the coordinates of the midpoint M. The x-coordinate of M is the average of the x-coordinates of P and Q, and the y-coordinate of M is the average of the y-coordinates of P and Q.
x-coordinate of M = (2 + 11) / 2 = 6.5
y-coordinate of M = (1 + 1) / 2 = 1
So the coordinates of M are (6.5, 1).
Now, let's find the length of MN using the distance formula:
√[(x2 - x1)^2 + (y2 - y1)^2]
= √[(6.5 - 8)^2 + (1 - 7)^2]
= √[(-1.5)^2 + (-6)^2]
= √[2.25 + 36]
= √38.25
≈ 6.18
Therefore, the length of MN is approximately 6.18 units.
Similarly, we can find the coordinates of the midpoint N using Q(11,1) and R(8,7), and then find the length of PQ using the distance formula.
x-coordinate of N = (11 + 8) / 2 = 9.5
y-coordinate of N = (1 + 7) / 2 = 4
So the coordinates of N are (9.5, 4).
Now, let's find the length of PQ using the distance formula:
√[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 11)^2 + (1 - 1)^2]
= √[(-9)^2 + 0^2]
= √[81 + 0]
= √81
= 9
Therefore, the length of PQ is 9 units.
Comparing the lengths of MN and PQ, we can see that MN is shorter than PQ. MN is approximately 6.18 units, while PQ is 9 units.