$$ A(2,3), B(5,4), C(6,7) $$ the $$ riangle A B C $$ is

Question

$$ A(2,3), B(5,4), C(6,7) $$ the $$ 	riangle A B C $$ is

UpStudy AI · Accepted Answer

Below is the step‐by‐step solution to classify the triangle with vertices $$ A(2,3) $$, $$ B(5,4) $$, and $$ C(6,7) $$.

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### 1. Compute the side lengths using the distance formula

The distance formula between two points $$ (x_1,y_1) $$ and $$ (x_2,y_2) $$ is given by:
$$
d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
$$

#### Side $$ AB $$:
$$
AB = \sqrt{(5-2)^2 + (4-3)^2} = \sqrt{3^2+1^2} = \sqrt{9+1} = \sqrt{10}
$$

#### Side $$ BC $$:
$$
BC = \sqrt{(6-5)^2 + (7-4)^2} = \sqrt{1^2+3^2} = \sqrt{1+9} = \sqrt{10}
$$

#### Side $$ AC $$:
$$
AC = \sqrt{(6-2)^2 + (7-3)^2} = \sqrt{4^2+4^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}
$$

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### 2. Analyze the side lengths

From the calculations:
- $$ AB = \sqrt{10} $$
- $$ BC = \sqrt{10} $$
- $$ AC = 4\sqrt{2} $$

Since $$ AB = BC $$ and $$ AC $$ is different, the triangle has **two equal sides**.

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### 3. Conclusion

Because the triangle has two sides of equal length, $$ \triangle ABC $$ is an **isosceles triangle**.

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### Final Answer

$$ \triangle ABC $$ is an **isosceles triangle**.
$$ \triangle ABC $$ is an isosceles triangle.

\( A(2,3), B(5,4), C(6,7) \) the \( \triangle A B C \) is