in triangle ABC orthocentre is (2, 3) and circumcentre is (5, -6) then the centroid is
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Let ABC be a triangle having orthocentre and circumcentre at (9 , 5) and (0 ,0 ) respectively. If the equation of side BC is 2x−y=10 ,then find the possible coordinates of vertex A.
December 26, 2019
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Neelima Lalani
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ANSWER
Co-ordinates of centro-id (G)≡(
2+1
2×0+9×1
,
2+1
2×0+5×1
)≡(
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
Co-ordinates of centro-id (G)≡ (3,
3
5
)≡(
3
x
1
+x
2
+x
3
,
3
y
1
+y
2
+y
3
)
Now,
⇒
3
x
1
+x
2
+x
3
=3
⇒ x
1
+x
2
+x
3
=9 ----- ( 1 )
⇒
3
y
1
+y
2
+y
3
=
3
5
⇒ y
1
+y
2
+y
3
=5 ----- ( 2 )
D(h,k) lie on the line BC, so it will satisfy the equation 2h−k=10
Now,
Slope of CD× Slope of BC=−1 [ Since, both are perpendicular to each other ]
⇒
h−0
k−0
×2=−1
⇒ h=−2k
Substituting value of h in given equation we get,
⇒ 2(−2k)−k=10
⇒ −4k−k=10
⇒ −5k=10
∴ k=−2
⇒ h=−2k=−2(−2)=4
So, we got D co-ordinates (4,−2)=(
2
x
2
+x
3
,
2
y
2
+y
3
)
⇒
2
x
2
+x
3
=4
⇒ x
2
+x
3
=8
Substituting above in ( 1 ) we get,
x
1
+8=9
∴ x
1
=1
⇒
2
y
2
+y
3
=−2
⇒ y
2
+y
3
=−4
Substituting above in ( 2 ) we get,
y
1
−4=5
∴ y
1
=9
∴ The possible co-ordinates of A is (1,9)