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?.
(its a triangle)
tryin to figure that one out too
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?.
PR
(2+8)/2,(1+7)/2
10 8
10/2=5
8/2=4
QR
(11+8)/2,(1+7)/2
19 8
19/2=9.5
8/2=4
PQ
To find the lengths MN and PQ, we can use the distance formula:
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Let's start by finding the coordinates of M and N, the midpoints of PQ and PR, respectively.
1. Midpoint M:
To find the midpoint of PQ (PQ), we add the x-coordinates and divide by 2, and add the y-coordinates and divide by 2.
x-coordinate of M = (x-coordinate of P + x-coordinate of Q) / 2
= (2 + 11) / 2 = 13 / 2 = 6.5
y-coordinate of M = (y-coordinate of P + y-coordinate of Q) / 2
= (1 + 1) / 2 = 2 / 2 = 1
Therefore, the coordinates of M are (6.5, 1).
2. Midpoint N:
To find the midpoint of PR (PN), we perform the same calculation as above.
x-coordinate of N = (x-coordinate of P + x-coordinate of R) / 2
= (2 + 8) / 2 = 10 / 2 = 5
y-coordinate of N = (y-coordinate of P + y-coordinate of R) / 2
= (1 + 7) / 2 = 8 / 2 = 4
Therefore, the coordinates of N are (5, 4).
Now, let's calculate the lengths MN and PQ using the distance formula.
Distance MN:
MN = √((x2 - x1)^2 + (y2 - y1)^2)
= √((5 - 6.5)^2 + (4 - 1)^2)
= √((-1.5)^2 + 3^2)
= √(2.25 + 9)
= √11.25
≈ 3.35 (rounded to two decimal places)
Distance PQ:
PQ = √((x2 - x1)^2 + (y2 - y1)^2)
= √((11 - 2)^2 + (1 - 1)^2)
= √(9^2 + 0)
= √(81 + 0)
= √81
= 9
Comparing the lengths:
MN ≈ 3.35
PQ = 9
MN is significantly shorter than PQ.