if the points (10,5) and (8,4) and (6,6) are mid points of a triangle find its vertices
Answers
Answer:
Given-
In ΔABC, the mid point of BC is D(8,4),
The mid point of AC is E(10,5), and the mid point of AB is F(6,6).
To find out -
The coordinates of A,B & C.
Solution-
Let the vertices of ΔABC be A(x
1
,y
1
),B(x
2
,y
2
) & C(x
3
,y
3
).
Then, by applying mid point formula,
2
x
2
+x
3
=8
⟹x
2
+x
3
=16 and
2
y
2
+y
3
=4
⟹y
2
+y
3
=8 ........(i)
2
x
1
+x
3
=10
⟹x
1
+x
3
=20 and
2
y
1
+y
3
=5
⟹y
1
+y
3
=10 ........(ii),
2
x
1
+x
2
=6
⟹x
1
+x
2
=12 and
2
y
1
+y
2
=6
⟹y
1
+y
2
=12 ........(iii).
Adding (i), (ii) & (iii),
2(x
1
+x
2
+x
3
)=48 and
2(y
1
+y
2
+y
3
)=30
⟹x
1
+x
2
+x
3
=24 and
y
1
+y
2
+y
3
=15 ...........(iv).
Subtracting (i) from (iv)
x
2
=24−20=4 and y
2
=15−10=5,
Subtracting (ii) from (iv)
x
1
=24−16=8 and y
1
=15−8=7,
Subtracting (iii) from (iv)
x
3
=24−12=12 and y
2
=15−12=3.
∴ The coordinates of the vertices are A(8,7),B(4,5) & C(12,3)