In `DeltaABC, AB = 8` cm, `BC=15` cm and `AC = 17` cm. find the largest angle.

Options

A. `angleA`
B. `angleC`
C. `angleB`
D. None of these

Answer

To find the largest angle in triangle ABC, we can use the property that the largest angle is opposite the longest side. Here are the steps to solve the problem:

### Step 1: Identify the lengths of the sides
We have the following sides:
- AB = 8 cm
- BC = 15 cm
- AC = 17 cm

### Step 2: Determine the longest side
Among the sides AB, BC, and AC, we can see that:
- AB = 8 cm
- BC = 15 cm
- AC = 17 cm

The longest side is AC, which measures 17 cm.

### Step 3: Identify the angle opposite the longest side
The angle opposite the longest side (AC) is angle B. Therefore, angle B is the largest angle in triangle ABC.

### Step 4: Use the cosine rule to find angle B
We can use the cosine rule to find the measure of angle B. The cosine rule states that:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where \( C \) is the angle opposite side \( c \).

In our case:
- \( a = AB = 8 \) cm
- \( b = BC = 15 \) cm
- \( c = AC = 17 \) cm

Plugging in the values:
\[ 17^2 = 8^2 + 15^2 - 2 \cdot 8 \cdot 15 \cdot \cos(B) \]

### Step 5: Calculate the squares of the sides
Calculating the squares:
- \( 17^2 = 289 \)
- \( 8^2 = 64 \)
- \( 15^2 = 225 \)

Now substitute these values into the equation:
\[ 289 = 64 + 225 - 2 \cdot 8 \cdot 15 \cdot \cos(B) \]

### Step 6: Simplify the equation
Combine the terms on the right:
\[ 289 = 289 - 240 \cdot \cos(B) \]

### Step 7: Isolate cos(B)
Subtract 289 from both sides:
\[ 0 = -240 \cdot \cos(B) \]

This simplifies to:
\[ 240 \cdot \cos(B) = 0 \]

### Step 8: Solve for cos(B)
Dividing both sides by 240 gives:
\[ \cos(B) = 0 \]

### Step 9: Find angle B
The angle for which the cosine is 0 is:
\[ B = 90^\circ \]

### Conclusion
Thus, the largest angle in triangle ABC is \( 90^\circ \).

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