show that the piont taken in onder form an isosceles triangle A(2,5) ,B(2,0),C(-2,3) RCN
Answers
Answer:
Let ΔABC be the isosceles triangle, the third vertex be C(a,b).
and A(2,0) and B(2,5)
Let AC and BC be equal sides of the triangle
By distance formula we have,
D=
(x
2
−x
1
)+(y
2
−y
1
)
∴AB=
(2−2)
2
+(5−0)
2
=
25
=5
Now,
AC=3 [length of equal sides=3 ]
AC
2
=3
2
=9..........(a)
but by distance formula,
AC
2
=
(a−2)
2
+b
2
...........(b)
Combining (a) and (b)
⇒(a−2)
2
+b
2
=9
⇒a
2
+4−4a+b
2
=9
⇒a
2
+b
2
−4a=5 ---------- (1)
Consider BC
BC
2
=
(a−2)
2
+(b−5)
2
but BC=3
∴(a−2)
2
+(b−5)
2
=9
⇒a
2
+4−4a+b
2
+25−10b=9
⇒a
2
+b
2
−4a−10b+20=0 ------- (2)
⇒−10b+20+5=0 ---------(from 1 )
⇒10b=25
⇒b=
10
25
=
2
5
Now, a
2
+
4
25
−4a=5
⇒a
2
−4a=
4
−5
⇒a
2
−4a+
4
5
=0
⇒a=
2
4±
16−4∗1∗
4
5
=
2
4±
11
=
2
2+
11
,
2
2−
11
∴ Coordinate of third vertex (a,b) = (
2
2+
11
,
2
5
) and (
2
2−
11
,
2
5
)