Question

Даны точки А(2;3), В(5;5), С(8;3) и D(5;1). докажите, что АС и ВD пересекаются и тоской пересечения делятся пополам."

Answer 1

М- точка пересечения
М(5;3)

Answer 2

Geometric Proof of Intersection and Midpoint Division

To prove that the line segments AC and BD intersect and that their point of intersection divides them in half, we can use the concept of midpoint and the properties of linear equations.

Step 1: Finding the Midpoint of AC

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula: \[ \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]

Using this formula, we can find the midpoint of AC: \[ \left( \frac{2 + 8}{2}, \frac{3 + 3}{2} \right) = (5, 3) \]

Step 2: Finding the Midpoint of BD

Similarly, the midpoint of BD can be found using the coordinates of B and D: \[ \left( \frac{5 + 5}{2}, \frac{5 + 1}{2} \right) = (5, 3) \]

Step 3: Showing Intersection

Now, to prove that AC and BD intersect, we can compare the slopes of the two line segments. If the slopes are different, the lines intersect.

The slope of AC is given by: \[ m_{AC} = \frac{3 - 3}{2 - 8} = 0 \]

The slope of BD is given by: \[ m_{BD} = \frac{1 - 5}{5 - 5} = undefined \]

Since the slopes are different, AC and BD intersect.

Step 4: Proving Midpoint Division

To prove that the point of intersection divides the line segments in half, we can use the midpoint formula again to find the coordinates of the point of intersection.

The coordinates of the point of intersection can be found by solving the equations of the lines AC and BD simultaneously.

The equation of AC is: \[ y = 3 \]

The equation of BD is: \[ x = 5 \]

The point of intersection is (5, 3), which is the midpoint of both line segments AC and BD.

Therefore, we have shown that AC and BD intersect at the point (5, 3) and that this point divides both line segments in half.