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The Perimeter of a Triangle is 50 cm. One Side of a Triangle is 4 cm Longer than the Smaller Side and the Third Side is 6 cm less than Twice the Smaller Side. Find the Area of the Triangle
by T Santhosh
Updated Jun 12, 2024
The Perimeter of a Triangle is 50 cm. One Side of a Triangle is 4 cm Longer than the Smaller Side and the Third Side is 6 cm less than Twice the Smaller Side. Find the Area of the Triangle
To find the area of the triangle with a given perimeter of 50 cm and specific side relationships.
Identify Side Lengths:
Let's denote the smallest side of the triangle as x cm.
The second side is then (x+4) cm long because it's 4 cm longer than the smallest side.
The third side is (2x−6) cm long, as it's 6 cm less than twice the smallest side.
Since the perimeter is 50 cm, we can write the equation
x+(x+4)+(2x−6) = 50 to represent the sum of all three sides.
Simplifying the equation 4x = 52 gives us x = 13 cm, which is the length of the smallest side.
With x = 13 cm, the sides of the triangle are 13 cm, 17 cm, and 20 cm.
The semi-perimeter, denoted as s, is half the sum of the sides
s = (13+17+20)/2 = 25 cm
Apply Heron's Formula for Area:
Using Heron's Formula for the area of a triangle,
Area = √s(s - a)(s - b)(s - c)
where a, b, and c are the sides of the triangle.
Therefore Area = √25x12x8x5 = 20√30 square centimeters.
Concept of Area of a Triangle
The area of a triangle is a fundamental concept in geometry. It represents the amount of space enclosed within the triangle's boundaries. Understanding how to calculate the area of a triangle is essential for various applications in mathematics.
The simplest formula for the area of a triangle is Area = 1/2 x base x height
This formula applies when you know the base of the triangle and its corresponding height (the perpendicular distance from the base to the opposite vertex).
Heron's Formula:
Heron's Formula is useful when you know the lengths of all three sides of the triangle but not the height. It states that the area can be calculated using the semi-perimeter (half the perimeter) of the triangle.
The formula is Area = √s(s - a)(s - b)(s - c)
where s is the semi-perimeter, and a, b, and c are the lengths of the sides.
Trigonometric Formula:
Another method involves using trigonometry, particularly when you know two sides and the included angle.
The formula is Area = 1/2 x a x b x sin(C)
where a and b are two sides, and C is the angle between them.
