Chassily $$ riangle A B C $$ by its sides. Then determine whether it is a right triangle. $$ A(3,3), B(6,9), C(6,-3) $$ The triangle can be classified by its sides as This triangle $$ \square $$ a right triangle.

Question

Chassily $$ 	riangle A B C $$ by its sides. Then determine whether it is a right triangle. $$ A(3,3), B(6,9), C(6,-3) $$ The triangle can be classified by its sides as This triangle $$ \square $$ a right triangle.

UpStudy AI · Accepted Answer

To determine whether triangle $$ ABC $$ is a right triangle, we can use the coordinates of the points $$ A(3,3) $$, $$ B(6,9) $$, and $$ C(6,-3) $$ to calculate the lengths of its sides.

1. **Calculate the lengths of the sides:**
   - The length of side $$ AB $$:
     $$
     AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(6 - 3)^2 + (9 - 3)^2}
     $$
   - The length of side $$ BC $$:
     $$
     BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(6 - 6)^2 + (-3 - 9)^2}
     $$
   - The length of side $$ CA $$:
     $$
     CA = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2} = \sqrt{(3 - 6)^2 + (3 - (-3))^2}
     $$

Now, let's calculate these lengths step by step.

### Step 1: Calculate $$ AB $$
$$
AB = \sqrt{(6 - 3)^2 + (9 - 3)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}
$$

### Step 2: Calculate $$ BC $$
$$
BC = \sqrt{(6 - 6)^2 + (-3 - 9)^2} = \sqrt{0 + (-12)^2} = \sqrt{144} = 12
$$

### Step 3: Calculate $$ CA $$
$$
CA = \sqrt{(3 - 6)^2 + (3 - (-3))^2} = \sqrt{(-3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45}
$$

Now we have the lengths of the sides:
- $$ AB = \sqrt{45} $$
- $$ BC = 12 $$
- $$ CA = \sqrt{45} $$

2. **Check if it is a right triangle:**
A triangle is a right triangle if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides.

Let's identify the longest side:
- The lengths are $$ \sqrt{45} $$, $$ 12 $$, and $$ \sqrt{45} $$. The longest side is $$ BC = 12 $$.

Now we check:
$$
AB^2 + CA^2 = (\sqrt{45})^2 + (\sqrt{45})^2 = 45 + 45 = 90
$$
$$
BC^2 = 12^2 = 144
$$

Since $$ AB^2 + CA^2 \neq BC^2 $$, triangle $$ ABC $$ is not a right triangle.

### Conclusion
The triangle can be classified by its sides as:
This triangle $$ \square $$ a right triangle.

The final answer is:
This triangle is **not** a right triangle.
This triangle is not a right triangle.

Chassily by its sides. Then determine whether it is a right triangle.

The triangle can be classified by its sides as
This triangle a right triangle.

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Chassily by its sides. Then determine whether it is a right triangle. The triangle can be classified by its sides as This triangle a right triangle.