Decrease in a Triangle’s Perimeter When Sides are Halved

Understanding the relationship between the sides and the perimeter of a triangle is essential in geometry. In this article, we explore what happens to the perimeter of a triangle when each of its sides is halved. We will also discuss the implications when the triangle is a right triangle.

Concept of Perimeter

The perimeter of a triangle is defined as the sum of the lengths of its three sides. If we denote the lengths of the sides by (a), (b), and (c), the original perimeter (P) can be expressed as:

[text{Original Perimeter, } P a b c]

Halving the Sides of a Triangle

When each side of the triangle is halved, the new lengths become (frac{a}{2}), (frac{b}{2}), and (frac{c}{2}). Thus, the new perimeter (P') is given by:

[text{New Perimeter, } P' frac{a}{2} frac{b}{2} frac{c}{2} frac{1}{2}(a b c)]

By simplifying, we find:

[text{New Perimeter, } P' frac{P}{2}]

Decrease in Perimeter

The decrease in the perimeter can be calculated as the difference between the original perimeter and the new perimeter:

[text{Decrease in Perimeter} P - P' P - frac{P}{2} frac{P}{2}]

Hence, the perimeter decreases by half when the sides of the triangle are halved.

Right Triangle and Halving Sides

Consider a right triangle, where one of the angles is 90 degrees. When the sides of this right triangle are halved, the triangle remains a right triangle, but the lengths of the legs and the hypotenuse are reduced by half. The perimeter of the original right triangle with sides (a), (b), and hypotenuse (c) is:

[text{Original Perimeter, } P a b c]

The new perimeter (P') after halving the sides is:

[text{New Perimeter, } P' frac{a}{2} frac{b}{2} frac{c}{2} frac{1}{2}(a b c)]

Again, simplifying the expression:

[text{New Perimeter, } P' frac{P}{2}]

Thus, the perimeter of the right triangle also decreases by half when its sides are halved.

Conclusion

In this article, we have explored the relationship between the original and new perimeters of a triangle when the sides are halved. Regardless of whether the triangle is right-angled or not, the perimeter always decreases by half. This principle is a fundamental concept in geometry and is crucial for understanding more complex geometric relationships.