Question

MNKL — прямоугольник со сторонами LK=16 см, NK = 26 см, а H — точка пересечения его диагоналей. Сколько точек пересечения имеет окружность с центром H и радиусом 6 см с прямой MN?

Answer 1

Ответ:

Диагонали прямоугольника равны и делятся пополам точкой пересечения. Так же это геометрический центр прямоугольника. Ширина треугольника - 26

Высота треугольника - 16

Разделим ширину на 2 - получим расстояние от H до стороны MN

26 ÷ 2 = 13 > 6 => точек пересечения окружности H с прямой MN нет.

Ответ: 0.

Answer 2

MNKL is a rectangle with sides LK = 16 cm and NK = 26 cm, and H is the point of intersection of its diagonals. The question asks how many points of intersection the circle with center H and radius 6 cm has with the line MN.

To solve this problem, we need to determine the position of the point H and then find the intersection points between the circle and the line MN.

Finding the position of point H:

To find the position of point H, we can use the fact that the diagonals of a rectangle bisect each other. This means that the point of intersection H is the midpoint of the diagonals.

The midpoint of the diagonals can be found by taking the average of the coordinates of the endpoints of the diagonals. In this case, the endpoints of the diagonals are the midpoints of the sides of the rectangle.

The midpoint of side LK can be found by taking half of the length of LK and adding it to the x-coordinate of point L. Similarly, the midpoint of side NK can be found by taking half of the length of NK and adding it to the x-coordinate of point N.

Let's calculate the coordinates of the midpoint of side LK and side NK:

Midpoint of side LK: x-coordinate: xL + (LK / 2) y-coordinate: yL

Midpoint of side NK: x-coordinate: xN + (NK / 2) y-coordinate: yN

Finding the intersection points between the circle and the line MN:

To find the intersection points between the circle with center H and radius 6 cm and the line MN, we can substitute the equation of the line MN into the equation of the circle and solve for the x-coordinates of the intersection points.

The equation of the line MN can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. Since the line MN is vertical, its slope is undefined.

Let's calculate the x-coordinates of the intersection points by substituting the equation of the line MN into the equation of the circle:

(x - xH)^2 + (y - yH)^2 = r^2

Substituting y = mx + b, we get:

(x - xH)^2 + (mx + b - yH)^2 = r^2

Expanding and simplifying the equation, we get a quadratic equation in terms of x:

(x^2 - 2xHx + xH^2) + (m^2x^2 + b^2 + yH^2 - 2mxyH - 2bxm + 2bxH) = r^2

Combining like terms, we get:

(1 + m^2)x^2 + (-2xH + 2mxyH - 2bxm + 2bxH) + (xH^2 + b^2 + yH^2 - 2yHb - r^2) = 0

This quadratic equation can be solved to find the x-coordinates of the intersection points. The number of intersection points depends on the discriminant of the quadratic equation.

If the discriminant is positive, there are two distinct real solutions, which means the line intersects the circle at two points. If the discriminant is zero, there is one real solution, which means the line is tangent to the circle. If the discriminant is negative, there are no real solutions, which means the line does not intersect the circle.

Let's calculate the discriminant and determine the number of intersection points.

Calculation:

Given: LK = 16 cm NK = 26 cm Radius of the circle = 6 cm

To find the coordinates of the midpoint of side LK, we need the coordinates of points L and K. Similarly, to find the coordinates of the midpoint of side NK, we need the coordinates of points N and K.

Unfortunately, the coordinates of points L, N, and K are not provided in the question. Without this information, we cannot accurately determine the position of point H or calculate the intersection points between the circle and the line MN.

Please provide the coordinates of points L, N, and K so that we can proceed with the calculation.