Understanding the Triangle and Its Properties

If you are presented with a triangle having side lengths of 15, 20, and 25, the first question may arise about the type of triangle this represents and how to find its altitudes. This article aims to delve into these concepts and pinpoint the shortest altitude in this specific triangle.

Identifying the Type of Triangle

This triangle is a right triangle, as suggested by its side lengths. A right triangle is one in which one of its internal angles is 90 degrees, making it a special case among triangles. The side lengths 15, 20, and 25 suggest that this is a scaled-up version of the 3-4-5 right triangle. In such triangles, the side opposite the right angle is the hypotenuse, which in this case is the longest side (25).

Step 1: Calculating the Area of the Triangle

The area of a right triangle can be calculated using the two shorter sides as the base and height. Here, the base can be considered as 15 and the height as 20. The formula for the area is:

$$text{Area} frac{1}{2} times text{base} times text{height} frac{1}{2} times 15 times 20 150$$

Step 2: Calculating the Altitudes of the Triangle

The altitude (or height) corresponding to each side of a triangle can be found using the formula:

$$text{Altitude} frac{2 times text{Area}}{text{side length}}$$

Applying this formula for each of the sides, we get:

Altitude to side 15: $$h_{15} frac{2 times 150}{15} frac{300}{15} 20$$ Altitude to side 20: $$h_{20} frac{2 times 150}{20} frac{300}{20} 15$$ Altitude to side 25 (hypotenuse): $$h_{25} frac{2 times 150}{25} frac{300}{25} 12$$

Step 3: Identifying the Shortest Altitude

By comparing the altitudes, we identify the shortest altitude as:

Altitude to side 15 is 20 Altitude to side 20 is 15 Altitude to side 25 (hypotenuse) is 12

Hence, the shortest altitude of this triangle is 12.

Alternative Methods and Further Exploration

The method described above can be applied to various types of triangles but it is also helpful to note that if it is a right triangle, the altitude from the right angle to the hypotenuse divides the triangle into two similar right triangles. This could provide a quicker way to solve for the altitudes.

Additionally, this problem can be solved using Heron's formula to calculate the area of the triangle and then determining the altitudes based on the area and side lengths. However, for this specific 15-20-25 triangle, the straightforward method of using the base-height area formula is efficient.

Conclusion

Understanding the properties of right triangles and how to calculate their altitudes is crucial in geometry. This exercise demonstrates how to find the shortest altitude in a right triangle by breaking down the problem into calculating the area and then using that area to find the altitudes. This approach is both systematic and mathematically solid, ensuring accurate calculations and deepening the understanding of triangle properties.