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

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.