The length of the triangle is in the ratio 3 : 4 : 5 the area of the triangle if the perimeter is 144cm.

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864 sq. cm

874 sq. cm

854 sq. cm

884 sq. cm

answer is A.

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Detailed Solution

We are given that the ratio of sides of the triangle is 3 : 4 : 5
Let us assume that from the figure of ΔABC
⇒ AB : BC : CA = 3 : 4 : 5
We know that the condition of ratios that if three numbers are in the ratio a : b : c then there exists an integer x such that the numbers are

a x, b x, c x

By using the above condition the sides of the triangle are given as
⇒AB = 3x
⇒BC = 4x
⇒CA = 5x
Let us assume that the area of the triangle as A
We know that the formula of area of a triangle that is
A = ½ (base) (height).
Here we can see that the square of CA is equal to the sum of squares of AB and BC that is

⇒ C A 2 = A B 2 + B C 2

We know that if the square of one side of a triangle is equal to the sum of squares of the other two sides then the triangle is a right-angled triangle and the two sides will be the base and height of the triangle.
By using the above condition we get the area of the triangle as
⇒ A = ½ (AB)(BC)……………….(1)
We are given that the perimeter of the triangle as 144 cm
We know that the perimeter of a triangle is given as the sum of sides of the triangle.
By using the above condition we have the perimeter of ΔABC as
⇒AB + BC + CA = 144
By substituting the required values in above equation we get
⇒3x + 4x + 5x = 144
⇒12x = 144
⇒x = 12
By substituting the value of x in the sides we get
⇒AB = 3×12 = 36
⇒ BC = 4 × 12 = 48
⇒CA = 5×12 = 60
By substituting the required sides in the equation (1) we get the area of the triangle as
⇒ A = ½ ×36×48
⇒ A = 864
Therefore we can conclude that the area of the given triangle is 864 sq. cm

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