M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 10, and PR = 15, then find OR.
M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 10, and PR = 15, then find OR.
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By the median theorem ON = QN / 3 = 10/3
NR = PR / 2 = 7.5
OR = sqrt{( 10/3)2 + 7.52} = 8.207381501
Note that NR =15/2 = 7.5 PN
Draw a perpendicular MS to PR
Then triangle PMS is similar to triangle PQN
Then since MP is 1/2 of PQ then PS = 1/2 of PN = 7.5/2= 15/4 = 3.75
And triangle PMS is similar to triangle PQN
So
PS/ PN = MS/QN
3.75/ 7.5 = MS /QN
1/2 = MS/10
MS = (1/2) 10 = 5
And triangle RSM is similar to triangle RNO
RS = 3.75 + 7.5 = 11.25
So
MS / RS = ON / NR
5/11.25 = ON / 7.5
(7.5)(5) / 11.25 = ON = 10/3
M is the midpoint of PQ and N is the midpoint of PR, and O is the intersection of QN and RM, as shown. If QN is perpendicular to PR, QN = 10, and PR = 15, then find OR.
-------------------------------------------------------------
By the median theorem ON = QN / 3 = 10/3
NR = PR / 2 = 7.5
OR = sqrt{( 10/3)2 + 7.52} = 8.207381501