To solve the problem of finding the coordinates of the other endpoint of a line segment given one endpoint and the angle of the segment, follow these steps:

Key Concepts

A line segment with angle $\alpha$ (measured from the positive x-axis) and length $L$ has a displacement vector from the known point to the unknown point given by:
$$(\Delta x, \Delta y) = (L \cdot \cos\alpha, L \cdot \sin\alpha)$$

General Formula

Let the known endpoint be $P(x_0, y_0)$, the unknown endpoint be $Q(x, y)$, the angle of the segment be $\alpha$, and the length be $L$. Then:
$$x = x_0 + L \cdot \cos\alpha$$
$$y = y_0 + L \cdot \sin\alpha$$

Example Application

Suppose:

• Known point: $(1, 2)$
• Angle $\alpha = 60^\circ$
• Length $L = 4$

Calculations:
$\cos60^\circ = 0.5$, $\sin60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$

$$x = 1 + 4 \cdot 0.5 = 3$$
$$y = 2 + 4 \cdot \frac{\sqrt{3}}{2} = 2 + 2\sqrt{3} \approx 5.464$$

Unknown endpoint: $(3, 2+2\sqrt{3})$

Answer:
For general values, use the formula above. For specific inputs (from the missing image), substitute the known values into the formula to get the exact coordinates.

If you provide the exact values (known point, angle, length), we can compute the numerical result directly.

$\boxed{(x_0 + L\cos\alpha, y_0 + L\sin\alpha)}$

(Note: Replace with numerical values once the image details are available.)
For example, if the known point is $(0,0)$, $\alpha=30^\circ$, $L=2$, the answer is $\boxed{(√3,1)}$.

Adjust based on your specific problem's parameters!

$\boxed{(x_0 + L\cos\alpha, y_0 + L\sin\alpha)}$

作者声明:本文包含人工智能生成内容。