The side lengths of △ABC are AB = 51, BC = 13, and AC = 40. What is the length of the
altitude from C to AB

To find the length of the altitude from \( C \) to \( AB \) in \( \triangle ABC \) where \( AB = 51 \), \( BC = 13 \), and \( AC = 40 \), we can use the area of the triangle and the formula for the altitude.

1. **Calculate the area of \( \triangle ABC \)**:
   We use Heron's formula for this:

   \[
   s = \frac{a + b + c}{2}
   \]
   where \( a = 51 \), \( b = 40 \), and \( c = 13 \).

   \[
   s = \frac{51 + 40 + 13}{2} = 52
   \]

   Now, the area \( \Delta \) is given by:

   \[
   \Delta = \sqrt{s(s - a)(s - b)(s - c)}
   \]

   Plugging in the values:

   \[
   \Delta = \sqrt{52(52 - 51)(52 - 40)(52 - 13)} = \sqrt{52 \cdot 1 \cdot 12 \cdot 39}
   \]

   Calculate inside the square root:

   \[
   \Delta = \sqrt{52 \cdot 1 \cdot 12 \cdot 39} = \sqrt{24336}
   \]

   \[
   \Delta = 156
   \]

2. **Find the altitude \( h \) from \( C \) to \( AB \)**:
   The area of the triangle can also be expressed as:

   \[
   \Delta = \frac{1}{2} \times \text{base} \times \text{height}
   \]

   Here, the base is \( AB = 51 \), and the height is the altitude \( h \) from \( C \) to \( AB \).

   \[
   156 = \frac{1}{2} \times 51 \times h
   \]

   Solving for \( h \):

   \[
   156 = 25.5 \times h
   \]

   \[
   h = \frac{156}{25.5} = 6.1176
   \]

Therefore, the length of the altitude from \( C \) to \( AB \) is approximately \( 6.12 \) units.

Would you like more details on this calculation or have any other questions? 

Here are some related questions you might be interested in:

1. How do you derive Heron's formula?
2. What is the significance of the semiperimeter in triangle calculations?
3. How would the calculation change if the side lengths were different?
4. What are other methods to find the area of a triangle?
5. How can you find the altitude if only given the coordinates of the vertices?
6. What role does the altitude play in triangle properties?
7. Can you explain why the altitude from C to AB is perpendicular to AB?
8. How does the Pythagorean theorem relate to finding altitudes in right triangles?

**Tip**: When working with triangles, always verify that the given side lengths form a valid triangle using the triangle inequality theorem.

Learn how to find the length of the altitude from point C to side AB in triangle ABC with side lengths AB = 51, BC = 13, and AC = 40. Detailed step-by-step solution using Heron's formula and the area formula for triangles.

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