three vertices of a triangle are (2, 2) ,(2 ,8) and(-6 ,2).prove that the equation of the median drawn from the vertex (2 ,2 ) is (3x+4y-14)=0
Answers
Answer:
Lets coordinate of a triangle are A(x
1
,y
1
),B(x
2
,y
2
),C(x
3
,y
3
)
Here A(0,1),B(2,3),C(3,5)
(a) Centroid of the triangle=(x
c
,y
c
)
x
c
=
3
x
1
+x
2
+x
3
,y
c
=
3
y
1
+y
2
+y
3
x
c
=
3
0+2+3
=
3
5
,y
c
=
3
1+3+5
=3
So the centroid is (
3
5
,3).
(b) To find out the circumcenter we have to solve any two bisector equations and find out the intersection points.
So, mid point of AB =(
2
0+2
,
2
1+3
)=F(1,2)
Slope of AB=
3−1
2−0
=1
Slope of the bisector is the negative reciprocal of the given slope.
So, the slope of the perpendicular bisector =−1
Equation of line with slope −1 and the coordinates F(1,2) is,
(y−2)=−1(x−1)
x+y=3………………(1)
Similarly, for AC
Mid point of AC =(
2
0+3
,
2
1+5
)=E(
2
3
,3)
Slope of AC=
3−0
5−1
=
3
4
Slope of the bisector is the negative reciprocal of the given slope.
So, the slope of the perpendicular bisector =−
4
3
Equation of line with slope −
4
3
and the coordinates E(
2
3
,3) is,
(y−3)=−
4
3
(x−
2
3
)
y−3=−
4
3
x+
8
9
4
3
x+y=
8
33
………………(2)
By solving equation (1) and (2),
(1)−(2)⇒
4
x
=3−
8
33
;x=−
2
9
Substitute the value of x in to (1)
−
2
9
−y=3
y=−
2
15
So the circumcenter is (−
2
9
,−
2
15
)
(c) Let perpendicular bisectors from A,B,C of triangle be F (on side AB), E (on side AC),D(on side BC)
We know, slope of AB=1
Slope of CF = Perpendicular slope of AB
=−
Slope of AB
1
=−1
The equation of CF is given as,
y−5=−1(x−3)
x+y=8 ………………(1)
Slope of AC=
3
4
Slope of AD = Perpendicular slope of BC
=−
Slope of BC
1
=−
4
3
The equation of BE is given as,
y−3=−
4
3
(x−2)
y−3=−
4
3
x+
2
3
4
3
x+y=
2
9
………………(2)
By solving equation (1) and (2),
(1)−(2)⇒
4
x
=8−
2
9
;x=14
Substitute the value of x in to (1)
14+y=8
y=−6
So the Orthocenter is (14,−6)
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