Step 1: Calculate the Third Side
To find the area of the triangle, we first need to determine the lengths of all three sides. Given:
- Perimeter = 180 cm
- Side 1 = 8 cm
- Side 2 = 18 cm
We can find the third side as follows:
- Third Side = Perimeter - (Side 1 + Side 2)
- Third Side = 180 cm - (8 cm + 18 cm) = 154 cm
Step 2: Calculate the Area Using Heron's Formula
Now, we can calculate the area of the triangle using Heron's formula, which is:
- Semi-perimeter (s) = Perimeter / 2 = 180 cm / 2 = 90 cm
Using the semi-perimeter, we can find the area (A):
- Area (A) = √(s(s - a)(s - b)(s - c))
- Where a = 8 cm, b = 18 cm, c = 154 cm
- Area (A) = √(90(90 - 8)(90 - 18)(90 - 154))
Calculating this gives us:
- Area (A) = √(90 * 82 * 72 * -64) = 0 (since one term is negative, it indicates an invalid triangle)
Step 3: Check Validity of Triangle
Since the calculated area is zero, it’s clear that the sides do not form a valid triangle as they violate the triangle inequality theorem.
Step 4: Calculate the Altitude Corresponding to the Shortest Side
The shortest side is 8 cm. Since this is not a valid triangle, we cannot calculate the altitude. However, if it were valid, the formula for the altitude (h) corresponding to the base (shortest side):
- h = (2 * Area) / base
In this case, since Area is 0, the altitude will also be 0.
Conclusion
The triangle with the given sides cannot exist, thus the area is 0 and the altitude corresponding to the shortest side is also 0.