Consider a `Delta ABC` and let a,b, and c denote the leghts of the sides opposite to vertices A,B and C, respectively. Suppose `a=2,b =3, c=4` and H be the orthocentre. Find `15(HA)^(2).`

Answer

To solve the problem, we need to find \( 15(HA)^2 \) where \( H \) is the orthocenter of triangle \( ABC \) with sides \( a = 2 \), \( b = 3 \), and \( c = 4 \).

### Step 1: Understand the relationship between sides and angles
In triangle \( ABC \), we can use the relationship between the sides and angles to find the lengths of the altitudes and subsequently the distance from the orthocenter to vertex \( A \).

### Step 2: Use the sine rule
Using the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

### Step 3: Calculate the circumradius \( R \)
We can use the formula for the circumradius \( R \):
\[ R = \frac{abc}{4K} \]
where \( K \) is the area of triangle \( ABC \). We can find \( K \) using Heron's formula:
\[ s = \frac{a + b + c}{2} = \frac{2 + 3 + 4}{2} = 4.5 \]
Now, calculate the area \( K \):
\[ K = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{4.5(4.5-2)(4.5-3)(4.5-4)} = \sqrt{4.5 \times 2.5 \times 1.5 \times 0.5} \]
Calculating this gives:
\[ K = \sqrt{4.5 \times 2.5 \times 1.5 \times 0.5} = \sqrt{8.4375} \approx 2.9047 \]

### Step 4: Substitute into the circumradius formula
Now substituting \( K \) into the formula for \( R \):
\[ R = \frac{2 \times 3 \times 4}{4 \times 2.9047} = \frac{24}{11.6188} \approx 2.065 \]

### Step 5: Find the angles using the cosine rule
Using the cosine rule, we can find \( \cos A \):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{3^2 + 4^2 - 2^2}{2 \times 3 \times 4} = \frac{9 + 16 - 4}{24} = \frac{21}{24} = \frac{7}{8} \]
Thus, \( \sin A = \sqrt{1 - \cos^2 A} = \sqrt{1 - \left(\frac{7}{8}\right)^2} = \sqrt{\frac{15}{64}} = \frac{\sqrt{15}}{8} \).

### Step 6: Calculate \( HA \)
The distance from the orthocenter \( H \) to vertex \( A \) can be calculated using:
\[ HA = 2R \cos A \]
Substituting the values:
\[ HA = 2 \times 2.065 \times \frac{7}{8} = 2.065 \times \frac{14}{8} = 2.065 \times 1.75 \approx 3.62875 \]

### Step 7: Calculate \( (HA)^2 \)
Now we calculate \( (HA)^2 \):
\[ (HA)^2 \approx (3.62875)^2 \approx 13.136 \]

### Step 8: Calculate \( 15(HA)^2 \)
Finally, we find:
\[ 15(HA)^2 \approx 15 \times 13.136 \approx 197.04 \]

### Final Answer
Thus, the value of \( 15(HA)^2 \) is approximately \( 197.04 \).

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