`{:("Column A"," ","Column B"),("Measure of the largest angle(in degree) of a triangle with sides of length 5,6 and 7",,60):}`

Answer

To find the measure of the largest angle in a triangle with sides of lengths 5, 6, and 7, we can use the Law of Cosines. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 5 \)
- \( b = 6 \)
- \( c = 7 \)

### Step 2: Determine which angle to find
To find the largest angle, we need to find the angle opposite the longest side, which is side \( c = 7 \). Therefore, we will find angle \( C \).

### Step 3: Apply the Law of Cosines
The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Substituting the values:
\[ 7^2 = 5^2 + 6^2 - 2 \cdot 5 \cdot 6 \cdot \cos(C) \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 49 = 25 + 36 - 60 \cdot \cos(C) \]
This simplifies to:
\[ 49 = 61 - 60 \cdot \cos(C) \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate \( \cos(C) \):
\[ 60 \cdot \cos(C) = 61 - 49 \]
\[ 60 \cdot \cos(C) = 12 \]
\[ \cos(C) = \frac{12}{60} = \frac{1}{5} \]

### Step 6: Find angle C
Now, we find angle \( C \) using the inverse cosine function:
\[ C = \cos^{-1}\left(\frac{1}{5}\right) \]

### Step 7: Calculate the angle
Using a calculator:
\[ C \approx 78.46^\circ \]

### Conclusion
Thus, the measure of the largest angle in the triangle with sides of lengths 5, 6, and 7 is approximately \( 78.46^\circ \).

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