Question

Circle C is inscribed in triangle QSU. Circle C is inscribed in triangle Q S U. Points R, T, and V of the circle are on the sides of the triangle. Point R is on side Q S, point T is on side S U, and point V is on side Q U. The length of Q R is 10, the length of R S is 2 x, the length of S T is x + 3, and the length of T U is 4. What is the perimeter of triangle QSU? 3 units 16 units 30 units 40 units

Answer 1

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