The altitudes \( h_a \), \( h_b \), and \( h_c \) corresponding to sides \( a = 13 \), \( b = 14 \), and \( c = 15 \) are calculated as follows:
["Understanding Triangle Heights: Calculating ( h_a ), ( h_b ), and ( h_c ) for Sides 13, 14, and 15", "In geometry, understanding the heights (or altitudes) of a triangle relative to its sides is essential for analyzing triangle properties, solving problems involving area, and applying trigonometric relationships. In this article, we explore how to compute the altitudes ( h_a ), ( h_b ), and ( h_c ) corresponding to the sides ( a = 13 ), ( b = 14 ), and ( c = 15 ). These altitudes are calculated using the triangle’s area and the side lengths, providing insight into the triangle’s geometry.", "---", "### What Are Altitudes in a Triangle?", "An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Each side of the triangle has a corresponding altitude, and the area of the triangle can be expressed as:", "[\n\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}\n]", "This formula connects the altitudes directly to the triangle’s area. Therefore, to find each altitude ( h_a ), ( h_b ), and ( h_c ), we first calculate the triangle’s area.", "---", "### Step 1: Compute the Triangle’s Area Using Heron’s Formula", "Given sides:\n[\na = 13, \quad b = 14, \quad c = 15\n]", "First, calculate the semi-perimeter ( s ):", "[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\n]", "Using Heron’s formula, the area ( A ) is:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}\n]", "[\nA = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Break down the multiplication:", "[\n21 \ imes 8 = 168, \quad 7 \ imes 6 = 42, \quad 168 \ imes 42 = 7056\n]", "[\nA = \sqrt{7056} = 84\n]", "So, the area of the triangle is ( 84 ) square units.", "---", "### Step 2: Calculate Each Altitude Using the Area Formula", "Now use ( A = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height} ) to solve for each altitude.", "#### Altitude ( h_a ): Corresponding to side ( a = 13 )", "[\nA = \frac{1}{2} \ imes a \ imes h_a \Rightarrow 84 = \frac{1}{2} \ imes 13 \ imes h_a\n]", "Multiply both sides by 2:", "[\n168 = 13 \ imes h_a \Rightarrow h_a = \frac{168}{13}\n]", "[\nh_a \approx 12.923\n]", "#### Altitude ( h_b ): Corresponding to side ( b = 14 )", "[\n84 = \frac{1}{2} \ imes 14 \ imes h_b \Rightarrow 84 = 7 \ imes h_b\n]", "[\nh_b = \frac{84}{7} = 12\n]", "#### Altitude ( h_c ): Corresponding to side ( c = 15 )", "[\n84 = \frac{1}{2} \ imes 15 \ imes h_c \Rightarrow 84 = 7.5 \ imes h_c\n]", "[\nh_c = \frac{84}{7.5} = \frac{840}{75} = \frac{168}{15} = \frac{56}{5} = 11.2\n]", "---", "### Summary of Altitudes", "| Side ( \ ext{length} ) | Altitude ( h ) (over 2) |\n|--------------------------|----------------------------|\n| ( a = 13 ) | ( \frac{168}{13} \approx 12.923 ) |\n| ( b = 14 ) | ( 12 ) |\n| ( c = 15 ) | ( 11.2 ) or ( \frac{56}{5} ) |", "---", "### Why These Altitudes Matter", "Understanding these altitudes helps in:", "- Computing the area in different forms efficiently.\n- Analyzing triangle stability and center balance (e.g., orthocenter).\n- Solving for unknown lengths or angles in related geometric problems.", "---", "### Final Thoughts", "The altitudes ( h_a ), ( h_b ), and ( h_c ) for sides 13, 14, and 15 are precisely derived using Heron’s area formula and the fundamental relation between base, height, and area. This clean calculation method exemplifies how triangle geometry elegantly combines algebra and geometry. Whether for academic study or practical application, knowing how to compute these altitudes strengthens your geometric toolkit.", "---", "Keywords:\ntriangle altitudes, ( h_a ), ( h_b ), ( h_c ), side 13, side 14, side 15, Heron’s formula, triangle area, geometry, altitude calculation, formula altitude triangle.", "---", "Improve your geometrical calculations and deepen your understanding of triangle properties by mastering how to find altitudes from side lengths."]
