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 point \( C \) to the side \( AB \) in \(\triangle ABC\), we can use the formula for the area of a triangle and relate it to the base and height.

### Step 1: Calculate the area of \(\triangle ABC\)
We can use Heron's formula to find the area of the triangle. The formula is given by:

\[
s = \frac{a + b + c}{2}
\]

where \( s \) is the semi-perimeter, and \( a \), \( b \), and \( c \) are the side lengths of the triangle.

Given:
- \( AB = c = 51 \)
- \( BC = a = 13 \)
- \( AC = b = 40 \)

The semi-perimeter \( s \) is:

\[
s = \frac{51 + 13 + 40}{2} = \frac{104}{2} = 52
\]

Now, use Heron's formula to calculate the area \( A \):

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

\[
A = \sqrt{52(52-13)(52-40)(52-51)} 
\]

\[
A = \sqrt{52 \times 39 \times 12 \times 1} 
\]

\[
A = \sqrt{24336} = 156 \text{ square units}
\]

### Step 2: Relate the area to the altitude
The area of the triangle can also be expressed in terms of the base \( AB \) and the height \( h_C \) from \( C \) to \( AB \):

\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times h_C
\]

Substituting the known values:

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

### Step 3: Solve for \( h_C \)
Now, solve for the altitude \( h_C \):

\[
h_C = \frac{2 \times 156}{51} = \frac{312}{51} \approx 6.12 \text{ units}
\]

### Final Answer:
The length of the altitude from point \( C \) to side \( AB \) is approximately **6.12 units**.

Would you like more details or have any other questions?

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Here are some related questions to further explore:

1. How can you find the altitude from point \( A \) to side \( BC \)?
2. What is the perimeter of \(\triangle ABC\)?
3. How do you determine if \(\triangle ABC\) is a right triangle?
4. How can you find the circumradius of \(\triangle ABC\)?
5. What is the length of the altitude from \( B \) to \( AC \)?
6. How can you find the inradius of \(\triangle ABC\)?
7. How would the problem change if \( AB \) were the longest side?
8. Can you find the centroid of \(\triangle ABC\) and its coordinates?

**Tip:** When solving problems involving triangles, try using multiple methods (like trigonometry or coordinate geometry) to cross-verify your results.

Learn how to find the altitude from point C to side AB in triangle ABC using Heron's formula and the area of a triangle formula. Understand step-by-step calculations with given side lengths: AB = 51, BC = 13, and AC = 40. Suitable for high school students.

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