step by step explanation
Answers
Step-by-step explanation:
Solution :-
In ∆ ABC ,
AC = 52 cm
BC = 48 cm
Let AB = X cm
In ∆ ABC , there is another triangle ADB
In ∆ ADB , angle ADB = 90°
=> ∆ADB is a right angled triangle
AB is the hypotenuse = X cm
BD = 16 cm
AD = 12 cm
We know that
Pythagoras Theorem:-
In a right angled triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides
AB² = AD² + BD²
=> X² = 12²+16²
=> X² = 144+256
=> X² = 400
=> X=±√400
=> X = ±20 cm
X acn not be negative since it is the length of the side
X = 20 cm
Therefore , AB = 20 cm
To find the area of the shaded region we have to find the area of ∆ABC and Subtract the area of ∆ ADB from it.
Area of the shaded region = Area of ∆ ABC - Area of ∆ ADB
Finding Area of ∆ ADB :-
Area of a right angled triangle = (1/2)ab sq.units
We have a = AD = 12 cm
and b = BD = 16 cm
Area of ∆ ADB = (1/2)×AD×BD
=> Area of ∆ADB = (1/2)×(12×16)
=> Area of ∆ ADB = 6×16
=> Area of ∆ ADB = 96 cm² -----(1)
Finding Area of ∆ ABC:-
We have
AB = 20 cm , BC = 48 cm , CA = 52 cm
Area of triangle by Heron's formula
∆ =√[S(S-a)(S-b)(S-c)] sq.units
Where , S = (a+b+c)/2 units
Let a = 20 cm , b= 48 cm , c = 52 cm
S = (20+48+52)/2
=> S = 120/2
=> S = 60 cm
Now
Area of ∆ ABC
=> ∆=√[60(60-20)(60-48)(60-52)]
=> ∆ = √(60×40×12×8)
=> ∆ =√(3×20×20×2×3×4×4×2)
=> ∆ = √[(20×20)×(3×3)×(2×2)×(4×4)]
=> ∆ = 20×3×2×4 cm²
=> ∆ = 480 cm²
Area of ∆ ABC = 480 cm²-------(2)
Area of the shaded region
=> Area of ∆ABC - Area of ∆ ADB
=> (2)-(1)
=>480-96
=> 384 cm²
Answer:-
Area of the shaded region for the given problem is 384 cm²
Used formulae:-
Pythagoras Theorem:-
"In a right angled triangle The square of the hypotenuse is equal to the sum of the squares of the other two sides".
- The opposite side of the right angle is the hypotenuse.
- Area of a right angled triangle = (1/2)ab sq.units
- Area of triangle by Heron's formula
- ∆ =√[S(S-a)(S-b)(S-c)] sq.units
- S = (a+b+c)/2
- a ,b,c are the three sides of the triangle