In `triangleABC`, if `a=13, b=14, c=15`, then `tan((A)/(2))=`

Options

A. `4`
B. `2`
C. `(1)/(4)`
D. `(1)/(2)`

Answer

To find \( \tan\left(\frac{A}{2}\right) \) in triangle \( ABC \) with sides \( a = 13 \), \( b = 14 \), and \( c = 15 \), we can use the half-angle formula for tangent:

\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \]

where \( s \) is the semi-perimeter of the triangle, calculated as:

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

### Step 1: Calculate the semi-perimeter \( s \)

\[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \]

**Hint:** The semi-perimeter is half the sum of the lengths of the sides of the triangle.

### Step 2: Substitute \( s \), \( b \), and \( c \) into the formula

Now we will substitute \( s = 21 \), \( b = 14 \), and \( c = 15 \) into the formula for \( \tan\left(\frac{A}{2}\right) \):

\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \]

### Step 3: Calculate \( (s-b) \) and \( (s-c) \)

\[ s - b = 21 - 14 = 7 \]
\[ s - c = 21 - 15 = 6 \]

**Hint:** Subtract the lengths of sides \( b \) and \( c \) from the semi-perimeter \( s \).

### Step 4: Calculate \( (s-a) \)

\[ s - a = 21 - 13 = 8 \]

**Hint:** Similarly, subtract the length of side \( a \) from the semi-perimeter \( s \).

### Step 5: Substitute the values into the formula

Now we substitute these values into the formula:

\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(7)(6)}{21(8)}} \]

### Step 6: Simplify the expression

Calculating the numerator and denominator:

\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{42}{168}} \]

Now simplify \( \frac{42}{168} \):

\[ \frac{42}{168} = \frac{1}{4} \]

So,

\[ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Final Answer

Thus, the value of \( \tan\left(\frac{A}{2}\right) \) is:

\[ \boxed{\frac{1}{2}} \]

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