X is an interior point of an equilateral triangle PQR. X is at equidistant from PQ,QR,RP, then what is the value of angle PXQ
Answers
Answer:
Given PQ:
3
x+y–6=0
D=(
2
3
,
2
3
)
r=1
Let C(x
1
,y
1
) be the incenter.
⟹CD=1 = perpendicular ditstance from C to PQ
⟹
(x
1
−
2
3
)
2
+(y
1
−
2
3
)
2
=1–(1)
3+1
∣
3
x
1
+y
1
−6∣
=1
⟹∣
3
x
1
+y
1
−6∣=2(2)
Solving (1) and (2)
We get x=
3
,y=1
C=(
3
,1)
Given PQR is equilateral
Let QR be y=m
1
x+c
1
RP be y
2
=m
2
x+c
2
All the sides are at angle 60
∘
to each other.
PQ=
3
x+y–6,m=−
3
tan60=
1+mm’
∣m–m’∣
⟹
3
=
1−
3
m’
∣m’+
3
∣
⟹(
3
–3m’)
2
=(m’+
3
)
2
⟹8m’
2
=8
3
m’
m’=0
3
Let m
2
=0,m
2
=
3
Distance from C to QR and RP=r=1
From PR
1
2
+0
2
∣1–0−c
1
∣
=1
⟹c
1
=0,PR is y=0
From QR
1
2
+3
2
∣1–3−c
2
∣
=1
⟹c
2
=0,QR is y=
3
x
He'll army ☺☺
Have a bangtastic day❤❤
Stay safe and keep smile
Can u be my friend if u don't mind ✌✌