In `Delta ABC ` if a= 16 , b= 24 and c = 20 then `cos (B/2)`

Options

A. `3/4`
B. `1/4`
C. `1/2`
D. `1/3`

Answer

To find \( \cos \left( \frac{B}{2} \right) \) in triangle \( ABC \) where \( a = 16 \), \( b = 24 \), and \( c = 20 \), we can use the formula:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{s(s-b)}{ac}} \]

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

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

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

Given:
- \( a = 16 \)
- \( b = 24 \)
- \( c = 20 \)

Calculate \( s \):

\[ s = \frac{16 + 24 + 20}{2} = \frac{60}{2} = 30 \]

### Step 2: Substitute values into the formula for \( \cos \left( \frac{B}{2} \right) \)

Now, substitute \( s \), \( b \), \( a \), and \( c \) into the formula:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{30(30 - 24)}{16 \cdot 20}} \]

### Step 3: Simplify the expression

Calculate \( 30 - 24 \):

\[ 30 - 24 = 6 \]

Now substitute this back into the equation:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{30 \cdot 6}{16 \cdot 20}} \]

Calculate \( 16 \cdot 20 \):

\[ 16 \cdot 20 = 320 \]

So the expression becomes:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{180}{320}} \]

### Step 4: Simplify the fraction

Now simplify \( \frac{180}{320} \):

\[ \frac{180}{320} = \frac{9}{16} \]

### Step 5: Take the square root

Now take the square root:

\[ \cos \left( \frac{B}{2} \right) = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Final Answer

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

\[ \cos \left( \frac{B}{2} \right) = \frac{3}{4} \]

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