The ratio of equal and unequal sides of an isosceles triangle is 3: 5. If its perimeter is 110cm, then its area is?
Answers
Area of the given triangle is 414.5 cm²
Given :-
The ratio of equal and unequal sides of an isosceles triangle is 3:5.
Its perimeter is 110cm.
To find :-
Area of the triangle .
Solution :-
Given that
The ratio of equal and unequal sides of an Isosceles triangle = 3:5
Let they be 3X cm and 5X cm
The length of equal sides = 3X cm each
The length of unequal side = 5X cm
We know that
The perimeter of an Isosceles triangle is 2a+b units
Where, a is the length of equal sides and b is the length of unequal side.
The perimeter of the Isosceles triangle
=> P = 2(3X)+5X cm
=> P = 6X+5X cm
=> P = 11X cm
According to the given problem
The perimeter of the triangle = 110 cm
Therefore, 11X = 110
=> X = 110/11
=> X = 10 cm
If X = 10 cm then 3X = 3(10) = 30 cm
If X = 10 cm then 5X = 5(10) = 50 cm
We have,
The lengths of the three sides of the triangle are 30 cm , 30 cm and 50 cm
We know that
Area of an Isosceles triangle
= (b/4)√(4a²-b²) sq.units
We have , a = 30 cm and b = 50 cm
Area of the given triangle
=> A = (50/4)√[4(30²)-50²] cm²
=> A = (25/2)√[4×900-2500]
=> A = (25/2)√(3600-2500)
=> A = (25/2)√1100
=> A = (25×10/2) √11
=> A = (250/2)√11
=> A = 125√11 cm² or
=> A = 125× 3.316
=> A = 414.5 cm²
(or)
we have, a = 30 cm , b = 30 cm ,
c = 50 cm
P = 110 cm, S = P/2 = 110/2 = 55 cm
We know that
Area of a triangle by Heron's formula
A = √[S(S-a)(S-b)(S-c)] sq.units
=> A = √[55(55-30(55-30)(55-50)] cm²
=> A = √[55(25)(25)5]
=> A = √(25×25×25×11)
=> A = 25×5√11
=> A = 125√11 cm² or
=> A = 125× 3.316
=> A = 414.5 cm²
Answer :-
The area of the given Isosceles triangle is 125√11 cm² or 414.5 cm²
Used formulae:-
• The perimeter of an Isosceles triangle is 2a+b units
• Area of an Isosceles triangle
= (b/4)√(4a²-b²) sq.units
- Where, a is the length of equal sides and b is the length of unequal side.
• Area of a triangle by Heron's formula
A = √[S(S-a)(S-b)(S-c)] sq.units
- a,b,c are the sides of the triangle.
- S = (a+b+c)/2 = P/2 units
• √11 = 3.316 ...
Answer:
Step-by-step explanation:
let the three sides be 3x,3x and 5x
therefore;
3x+3x+5x=110
11x=110
x=10
Hence the sides are30,30 ,50 cm