. O is the origin. B( −6, 9 ) and C ( 12, −3) are the vertices of the ∆ABC. If the  point P divides OB in the ratio 1 ∶ 2 and the point Q divides OC in the ratio  1 ∶ 2. Show that PQ =1 3 BC

Asked by pandurangaiah02 | 18th May, 2021, 12:01: PM

Expert Answer:

Given: O(0, 0), B(-6, 9), C(12, -3)
As P divides OB in the ratio 1 : 2, we have

straight P identical to open parentheses fraction numerator 1 cross times open parentheses negative 6 close parentheses plus 2 cross times 0 over denominator 1 plus 2 end fraction comma fraction numerator 1 cross times 9 plus 2 cross times 0 over denominator 1 plus 2 end fraction close parentheses equals open parentheses negative 2 comma space 3 close parentheses
As space straight Q space divides space OC space in space the space ratio space 1 space colon space 2 comma space we space have
straight Q identical to open parentheses fraction numerator 1 cross times 12 plus 2 cross times 0 over denominator 1 plus 2 end fraction comma fraction numerator 1 cross times open parentheses negative 3 close parentheses plus 2 cross times 0 over denominator 1 plus 2 end fraction close parentheses equals left parenthesis 4 comma space minus 1 right parenthesis
PQ equals square root of open parentheses negative 2 minus 4 close parentheses squared plus open parentheses 3 plus 1 close parentheses squared end root equals square root of 36 plus 16 end root equals square root of 52 equals 2 square root of 13
BC equals square root of open parentheses negative 6 minus 12 close parentheses squared plus open parentheses 9 plus 3 close parentheses squared end root equals square root of 324 plus 144 end root equals square root of 468 equals 6 square root of 13
rightwards double arrow PQ equals 1 third BC

Answered by Renu Varma | 21st May, 2021, 11:54: AM