vertices abc of triangle ABC and centres are drawn with radius 5 cm if ab is equal to 14 cm BC is equal to 48 cm CA is equal to 58 cm find the area of shaded region
Answers
Step-by-step explanation:
Given :-
In ∆ ABC, AB = 14 cm , BC = 48 cm and
CA = 58 cm and The radius of the arcs = 5 cm
To find :-
Find the area of the shaded region ?
Solution :-
Given that
In ∆ ABC,
AB = 14 cm
BC = 48 cm
CA = 58 cm
Vertices A,B,C of ∆ABC and centres are drawn with radius 5 cm
We know that
Area of a triangle whose sides are a,b and c units by Heron's formula is
∆ = √[S(S-a)(S-b)(S-c)] sq.units
Where, S = (a+b+c)/2 units
We have,
a = 14 cm
b = 48 cm
c = 58 cm
S = (14+48+58)/2 cm
=> S = 120/2 cm
=> S = 60 cm
Now
Area of ∆ABC = √[60(60-14)(60-48)(60-58)] sq.cm
=> Ar(∆ABC) =√(60×46×12×2) sq.cm
=> Ar(∆ABC) = √(4×5×3×23×2×4×3×2)
=> Ar(∆ABC) = √[(4×4)(2×2)(3×3)(5×23)]
=> Ar(∆ABC) = 4×2×3√115
=> Ar(∆ABC) = 24√115 sq.cm
=> Ar(∆ABC) = 24×10.72 sq.cm
=> Ar(∆ABC) = 257.28 sq.cm
Now
radius = 5 cm
We know that
Area of a sector = (x°/360°)×πr² sq.cm
Area of the three sectors
=> [(x1/360°)×πr² ]+ [(x1/360°)×πr²] +[(x1/360°)×πr² ]
=> (πr²/360°)(x1+x2+x3)
=>(πr²/360°)×180°
Since the sum of the three angles is equal to 180°
=> (πr²/2) sq.cm
=> [(22/7)×5²]/2
=> [(22/7)×25]/2
=> (22×25)/(7×2)
=> (11×25)/7
=> 275/7 sq.cm
=> 39.28 sq.cm
Now
Area of the shaded region =
Area of a triangle - Area of the three sectors
=> 257.28 -39.28 sq.cm
=> 218 sq.cm
Answer:-
Area of the shaded region is 218 sq.cm
Used formulae:-
→ Area of a triangle whose sides are a,b and c units by Heron's formula is
∆ = √[S(S-a)(S-b)(S-c)] sq.units
Where, S = (a+b+c)/2 units
→ The sum of the three angles in a triangle is equal to 180°
→ Area of a sector = (x°/360°)×πr² sq.cm
→ r = radius
→ x° = Angles subtend byan arc at the centre
→ π = 22/7