Find the tangents of the angles in triangle with vertices A(1, 1), B(5,2) and C(3, 5).


To find the tangents of the angles in a triangle with vertices \(A(1,1)\), \(B(5,2)\), and \(C(3,5)\), we can follow these steps:

### Step 1: Calculate the lengths of the sides of the triangle
We'll use the distance formula to calculate the lengths of each side of the triangle:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Side \(AB\):
\[
AB = \sqrt{(5 - 1)^2 + (2 - 1)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}
\]

#### Side \(BC\):
\[
BC = \sqrt{(3 - 5)^2 + (5 - 2)^2} = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

#### Side \(CA\):
\[
CA = \sqrt{(3 - 1)^2 + (5 - 1)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20}
\]

### Step 2: Use the tangent formula for the angles
The formula for the tangent of an angle in a triangle can be derived from the Law of Cosines or using the following trigonometric relation:
\[
\tan\left(\frac{\theta}{2}\right) = \frac{r}{s}
\]
However, using a more straightforward relation for the angles in the triangle:

- For \( \angle A \), the tangent can be expressed as:
\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}}


Find the tangents of the angles in triangle with vertices A(1, 1), B(5,2) and C(3, 5).

Learn how to calculate the tangents of the angles in a triangle with vertices at A(1, 1), B(5, 2), and C(3, 5). This problem uses geometry and trigonometry, including the distance formula and the tangent of an angle in a triangle. Suitable for high school students (Grades 9-12).

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