Question

1. На координатной плоскости отметьте точки А(-2;2), B(3;-5), C(-3; 2), Д6;0), Е (5:4) Найдите: а) координаты точки пересечения отрезка AB с осью ординат; б) координаты точки пересечения отрезка АС 4 осью абсцисс; в) координаты точки пересечения отрезков BE и СД; г) координату точки пересечения отрезков АД и СВ. образованных при

Answer 1

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Answer 2

Points on the Coordinate Plane

To solve this problem, we need to plot the given points on a coordinate plane and then find the coordinates of the requested intersections.

The given points are: - A(-2, 2) - B(3, -5) - C(-3, 2) - D(6, 0) - E(5, 4)

Let's plot these points on a coordinate plane:

A(-2, 2)   B(3, -5)   C(-3, 2)   D(6, 0)   E(5, 4)

a) Intersection of Segment AB with the y-axis

To find the coordinates of the intersection of segment AB with the y-axis (the ordinate axis), we need to find the point where the segment AB crosses the y-axis.

Since the y-axis is vertical and has an x-coordinate of 0, we can find the y-coordinate of the intersection point by finding the equation of the line passing through points A and B and substituting x = 0.

The equation of the line passing through two points (x1, y1) and (x2, y2) is given by:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

Substituting the coordinates of points A and B, we get:

y - 2 = (-5 - 2) / (3 - (-2)) * (0 - (-2))

Simplifying the equation gives us:

y - 2 = -7/5 * 2
y - 2 = -14/5
y = -14/5 + 2
y = -14/5 + 10/5
y = -4/5

Therefore, the coordinates of the intersection of segment AB with the y-axis are (0, -4/5).

б) Intersection of Segment AC with the x-axis

To find the coordinates of the intersection of segment AC with the x-axis (the abscissa axis), we need to find the point where the segment AC crosses the x-axis.

Since the x-axis is horizontal and has a y-coordinate of 0, we can find the x-coordinate of the intersection point by finding the equation of the line passing through points A and C and substituting y = 0.

The equation of the line passing through two points (x1, y1) and (x2, y2) is given by:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

Substituting the coordinates of points A and C, we get:

0 - 2 = (2 - 2) / (-3 - (-2)) * (x - (-2))
0 - 2 = 0 * (x + 2)
-2 = 0

Since the equation -2 = 0 is not true, it means that segment AC does not intersect the x-axis. Therefore, there is no point of intersection to find.

в) Intersection of Segments BE and CD

To find the coordinates of the intersection of segments BE and CD, we need to find the point where these two segments intersect.

To do this, we can find the equations of the lines passing through segments BE and CD and then solve the system of equations to find the point of intersection.

The equation of the line passing through two points (x1, y1) and (x2, y2) is given by:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

For segment BE, the equation of the line passing through points B and E is:

y - (-5) = (4 - (-5)) / (5 - 3) * (x - 3)
y + 5 = 9/2 * (x - 3)
y + 5 = 9/2 * x - 27/2
y = 9/2 * x - 27/2 - 5
y = 9/2 * x - 27/2 - 10/2
y = 9/2 * x - 37/2

For segment CD, the equation of the line passing through points C and D is:

y - 2 = (0 - 2) / (6 - (-3)) * (x - (-3))
y - 2 = -2/9 * (x + 3)
y = -2/9 * x - 2/9 * 3 + 2
y = -2/9 * x - 6/9 + 18/9
y = -2/9 * x + 12/9
y = -2/9 * x + 4/3

Now, we can solve the system of equations:

9/2 * x - 37/2 = -2/9 * x + 4/3

Simplifying the equation gives us:

81/2 * x - 37/2 = -4/9 * x + 4/3

Multiplying both sides of the equation by 2 and 9 to eliminate the fractions gives us:

81 * 9 * x - 37 * 9 = -4 * 2 * x + 4 * 2 * 3

Simplifying the equation gives us:

729 * x - 333 = -8 * x + 24

Moving all the terms to one side of the equation gives us:

729 * x + 8 * x = 333 + 24
737 * x = 357

Dividing both sides of the equation by 737 gives us:

x = 357 / 737

Therefore, the x-coordinate of the intersection point is approximately 0.485.

Substituting this value back into one of the equations, we can find the y-coordinate:

y = -2/9 * x + 4/3
y = -2/9 * 0.485 + 4/3
y = -0.107 + 1.333
y = 1.226

Therefore, the coordinates of the intersection of segments BE and CD are approximately (0.485, 1.226).

г) Intersection of Segments AD and BV

To find the coordinates of the intersection of segments AD and BV, we need to find the point where these two segments intersect.

To do this, we can find the equations of the lines passing through segments AD and BV and then solve the system of equations to find the point of intersection.

The equation of the line passing through two points (x1, y1) and (x2, y2) is given by:

y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)

For segment AD, the equation of the line passing through points A and D is:

y - 2 = (0 - 2) / (6 - (-2)) * (x - (-2))
y - 2 = -2/8 * (x + 2)
y = -2/8 * x - 2/8 * 2 + 2
y = -2/8 * x - 4/8 + 16/8
y = -2/8 * x + 12/8
y = -1/4 * x + 3/2

For segment BV, the equation of the line passing through points B and V is:

y - (-5) = (4 - (-5)) / (3 - 5) * (x - 5)
y + 5 = 9/(-2) * (x - 5)
y + 5 = -9/2 * (x - 5)
y + 5 = -9/2 * x + 45/2
y = -9/2 * x + 45/2 - 5
y = -9/2 * x + 45/2 - 10/2
y = -9/2 * x + 35/2

Now, we can solve the system of equations:

-1/4 * x + 3/2 = -9/2 * x + 35/2

Simplifying the equation gives us:

-2 * x + 6 = 18 * x - 70

Moving all the terms to one side of the equation gives us:

18 * x + 2 * x = 70 + 6

Simplifying the equation gives us:

20 * x = 76

Dividing both sides of the equation by 20 gives us:

x = 76 / 20

Therefore, the x-coordinate of the intersection point is approximately 3.8.

Substituting this value back into one of the equations, we can find the y-coordinate:

y = -1/4 * x + 3/2
y = -1/4 * 3.8 + 3/2
y = -0.95 + 1.5
y = 0.55

Therefore, the coordinates of the intersection of segments AD and BV are approximately (3.8, 0.55).

In summary, the coordinates of the requested intersections