Do the points (4, 3), (-4, -6) and (7, 9) form a triangle ? If yes, then find the longest side of the triangle.

Options

A. 18.6
B. 16.5
C. 24
D. 34

Answer

To determine whether the points (4, 3), (-4, -6), and (7, 9) form a triangle, we will calculate the distances between each pair of points and then check the triangle inequality theorem.

### Step 1: Calculate the distances between the points

1. **Distance between A (4, 3) and B (-4, -6)**:
\[ d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d_{AB} = \sqrt{((-4) - 4)^2 + ((-6) - 3)^2} = \sqrt{(-8)^2 + (-9)^2} = \sqrt{64 + 81} = \sqrt{145} \]

2. **Distance between B (-4, -6) and C (7, 9)**:
\[ d_{BC} = \sqrt{(7 - (-4))^2 + (9 - (-6))^2} \]
\[ d_{BC} = \sqrt{(7 + 4)^2 + (9 + 6)^2} = \sqrt{11^2 + 15^2} = \sqrt{121 + 225} = \sqrt{346} \]

3. **Distance between C (7, 9) and A (4, 3)**:
\[ d_{CA} = \sqrt{(4 - 7)^2 + (3 - 9)^2} \]
\[ d_{CA} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \]

### Step 2: Check the triangle inequality theorem

The triangle inequality theorem states that for three sides a, b, and c, the following must hold true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

Let:
- \( a = d_{AB} = \sqrt{145} \)
- \( b = d_{BC} = \sqrt{346} \)
- \( c = d_{CA} = \sqrt{45} \)

Now we check:
1. \( \sqrt{145} + \sqrt{346} > \sqrt{45} \)
2. \( \sqrt{145} + \sqrt{45} > \sqrt{346} \)
3. \( \sqrt{346} + \sqrt{45} > \sqrt{145} \)

### Step 3: Calculate the approximate values

1. \( \sqrt{145} \approx 12.04 \)
2. \( \sqrt{346} \approx 18.6 \)
3. \( \sqrt{45} \approx 6.71 \)

Now check the inequalities:
1. \( 12.04 + 18.6 > 6.71 \) → True
2. \( 12.04 + 6.71 > 18.6 \) → True
3. \( 18.6 + 6.71 > 12.04 \) → True

Since all inequalities hold true, the points do form a triangle.

### Step 4: Find the longest side

Now, we need to find the longest side among \( d_{AB} \), \( d_{BC} \), and \( d_{CA} \):
- \( d_{AB} = \sqrt{145} \approx 12.04 \)
- \( d_{BC} = \sqrt{346} \approx 18.6 \)
- \( d_{CA} = \sqrt{45} \approx 6.71 \)

The longest side is \( d_{BC} = \sqrt{346} \).

### Final Answer:
Yes, the points (4, 3), (-4, -6), and (7, 9) form a triangle, and the longest side of the triangle is \( \sqrt{346} \).

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