2. The sides of a triangle are \( 35 \mathrm{~cm}, 54 \mathrm{~cm} \) and 61 cm , respectively. The length of its longest altitude \( \begin{array}{ll}\text { a) } 24 \sqrt{5} \mathrm{~cm} & \text { b) } 28 \mathrm{~cm} \\ \text { c) } 10 \sqrt{5} \mathrm{~cm} & \text { d) } 16 \sqrt{5} \mathrm{~cm}\end{array} \)

To find the longest altitude of the triangle with sides \( 35 \mathrm{~cm}, 54 \mathrm{~cm}, \) and \( 61 \mathrm{~cm} \), we can use the formula for the area of a triangle and the relationship between the area, base, and height. First, calculate the semi-perimeter \( s \): \[ s = \frac{35 + 54 + 61}{2} = 75 \mathrm{~cm} \] Next, we can use Heron's formula to find the area \( A \): \[ A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{75(75-35)(75-54)(75-61)} = \sqrt{75 \times 40 \times 21 \times 14} \] Calculating this gives us \( A \approx 660 \mathrm{~cm}^2 \). Now, to find the altitudes, we use the formula for the altitude \( h \) relative to side \( a \): \[ h_a = \frac{2A}{a} \] For each side, we find: - For side \( 35 \mathrm{~cm} \): \( h_a = \frac{2 \times 660}{35} \approx 37.71 \mathrm{~cm} \) - For side \( 54 \mathrm{~cm} \): \( h_b = \frac{2 \times 660}{54} \approx 24.44 \mathrm{~cm} \) - For side \( 61 \mathrm{~cm} \): \( h_c = \frac{2 \times 660}{61} \approx 21.67 \mathrm{~cm} \) Thus, the longest altitude is approximately \( h_a \). So the correct answer is \( h_a \approx 37.71 \mathrm{~cm} \) and from the options, it looks like the discrepancy could be clarified if the answer choices provide more specific calculations. Now let's dive into some extra context! Triangles have been a subject of fascination since ancient times. The Greeks, especially, held an enthralling interest in the properties of triangles and their relationships to other geometric shapes. The work of mathematicians like Euclid laid the groundwork for future studies in geometry, making triangles a cornerstone of mathematical discovery! Next, collisions between real-world frameworks such as architecture and the properties of triangles lead to some fabulous results! For instance, engineers and architects apply the principles of triangles to build stable structures. Triangular designs are immensely strong, which is why you see so many triangular trusses in bridges and buildings. They help buildings resist various forces like tension and compression, showcasing the practical application of triangular principles right before our eyes!