The problem reads:

"Calculate the area of the triangle whose sides are 18 cm, 24 cm, and 30 cm in length. Also, find the length of the altitude corresponding to the smallest side."

Step 1: Verifying if the triangle is valid and a right triangle

The sides given are $a = 18 \, \text{cm}, b = 24 \, \text{cm}, \text{and } c = 30 \, \text{cm}$.

• Using the Pythagorean theorem: $c^2 = a^2 + b^2$: $30^2 = 18^2 + 24^2$ $900 = 324 + 576$ $900 = 900$ This confirms the triangle is a right triangle, with $c = 30 \, \text{cm}$ as the hypotenuse.

Step 2: Calculate the area of the triangle

The area of a right triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base and height are $18 \, \text{cm}$ and $24 \, \text{cm}$, respectively. Thus: $\text{Area} = \frac{1}{2} \times 18 \times 24 = \frac{1}{2} \times 432 = 216 \, \text{cm}^2$

Step 3: Find the altitude corresponding to the smallest side

The smallest side is $a = 18 \, \text{cm}$. Let the altitude corresponding to this side be $h$. The area of the triangle can also be expressed as: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{altitude}$ Substitute the known values: $216 = \frac{1}{2} \times 18 \times h$ $216 = 9h$ $h = \frac{216}{9} = 24 \, \text{cm}$

Final Answers:

1. Area of the triangle: $216 \, \text{cm}^2$
2. Altitude corresponding to the smallest side: $24 \, \text{cm}$

Would you like further explanation or additional details?

Related Questions:

1. What is Heron’s formula, and how can it be used for non-right triangles?
2. How do you verify if three sides form a valid triangle?
3. What are some properties of right triangles related to their altitudes?
4. How is the altitude related to the circumradius of the triangle?
5. Can this triangle be inscribed in a circle? If so, how do you find the radius?

Tip:

When working with right triangles, always identify the hypotenuse first—it simplifies calculations and checks!