10,7,8 are sides of triangle . Find projection of side of length '10' on '7' is

Answer

To find the projection of the side of length '10' on the side of length '7' in a triangle with sides 10, 7, and 8, we can follow these steps:

### Step 1: Identify the triangle and its sides
We have a triangle with sides:
- \( a = 10 \)
- \( b = 7 \)
- \( c = 8 \)

### Step 2: Use the Cosine Rule to find the angle between sides 10 and 7
To find the angle \( \theta \) between sides \( a \) (10) and \( b \) (7), we can use the Cosine Rule:
\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]
Substituting the values:
\[ 8^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \cos(\theta) \]
Calculating:
\[ 64 = 100 + 49 - 140 \cos(\theta) \]
\[ 64 = 149 - 140 \cos(\theta) \]
Rearranging gives:
\[ 140 \cos(\theta) = 149 - 64 \]
\[ 140 \cos(\theta) = 85 \]
\[ \cos(\theta) = \frac{85}{140} = \frac{17}{28} \]

### Step 3: Calculate the projection of side 10 on side 7
The projection of side \( a \) (10) onto side \( b \) (7) can be calculated using the formula:
\[ \text{Projection} = a \cdot \cos(\theta) \]
Substituting the values:
\[ \text{Projection} = 10 \cdot \frac{17}{28} \]
Calculating this gives:
\[ \text{Projection} = \frac{170}{28} = \frac{85}{14} \]

### Final Answer
The projection of the side of length 10 on the side of length 7 is \( \frac{85}{14} \).

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