Question

Диагональ квадрата расположена на прямой y=2x-1. Одна из его вершин находиться в точке в точке (4;-3). Составить уравнения его сторон и второй диагонали.

Answer 1

Даны прямая y=2x - 1 (диагональ d1 квадрата) и точка (4; -3).

Эта точка не лежит на  заданной прямой (2*4 - 1 ≠ -3).

Пусть это точка D (4; -3).

На перпендикуляре из точки D на заданную диагональ будет находиться вторая диагональ d2 квадрата.

Угловой коэффициент прямой d2 равен: k(d2) = -1/(k(d1)) = -1/2.

Уравнение d2: у = (-1/2)х + b. Для определения параметра d подставим координаты точки D: -3 = (-1/2)*4 + b. Отсюда b = -3 + 2 = -1.

Получили уравнение второй диагонали d2:

y = (-1/2)x - 1.

Далее используем формулу суммы тангенсов углов наклона прямой к оси Ох, так как этот тангенс является угловым коэффициентом прямой.

Угол между диагональю и стороной равен 45 градусов, его тангенс равен 1.

k(CD) = (2 + 1) / (1 - 2*1) = 3(-1) = -3.

Уравнение CD: y = -3x + b, для определения b подставим координаты точки D, через которую проходит прямая.

-3 = -3*4 + b, b = - 3+ 12 = 9.

Уравнение стороны CD: y = -3x + 9.

Пересечением диагонали и стороны CD найдём координаты точки С.

2х - 1 = -3х + 9, 5х = 10, х = 10/5 = 2, у = 2*2 - 1 = 3.

Точка С(2; 3).

Находим угловые коэффициенты уравнений двух других сторон квадрата как перпендикуляров.

k(BC) = k(AD) = -1/(-3) = 1/3.

Уравнение ВС: у = (1/3)х + b, подставим координаты найденной точки С.

3 = (1/3)*2 + b, b = 3 - (2/3) = 7/3.

Уравнение стороны BC: y = (1/3)x + (7/3).

Определяем уравнение стороны AD, у которой угловой коэффициент равен таковому стороны ВС.

Уравнение AD: у = (1/3)х + b, подставим координаты заданной точки D.

-3 = (1/3)*4 + b, b = -3 - (4/3) = -13/3.

Уравнение стороны  AD: y = (1/3 - (13/3).

Определим координаты точки В пересечением диагонали d2 и стороны ВС.

(-1/2)х - 1 = (1/3)х + (7/3), (5/6)х = -10/3, х = (-10/3)/(5/6) = -4,

у =(-1/2)*(-4) - 1 = 1.

Точка B(-4; 1).

Определяем уравнение стороны AВ, у которой угловой коэффициент равен таковому стороны CD.

Уравнение AB: у = -3х + b, подставим координаты найденной точки B.

1 = -3*(-4) + b, b = 1 - 12 = -11.

Уравнение стороны AB: y = -3x - 11.

Answer 2

Given Information

We are given that the diagonal of a square is located on the line y = 2x - 1, and one of its vertices is at the point (4, -3).

Finding the Equations of the Sides

To find the equations of the sides of the square, we need to determine the coordinates of the other three vertices.

Let's start by finding the slope of the line y = 2x - 1. The slope-intercept form of a line is y = mx + b, where m is the slope. In this case, the slope is 2.

Since the diagonal of the square is a line segment connecting two opposite vertices, it will have a slope perpendicular to the line y = 2x - 1. The negative reciprocal of 2 is -1/2, so the slope of the diagonal is -1/2.

Now, let's find the coordinates of the other three vertices.

1. Vertex 1: Given as (4, -3). 2. Vertex 2: To find this vertex, we can use the midpoint formula. The midpoint of the diagonal will be the midpoint of the line segment connecting Vertex 1 and Vertex 3 (opposite vertex). The midpoint formula is given by:

   x = (x1 + x2) / 2
   y = (y1 + y2) / 2
   

Substituting the values of Vertex 1 as (4, -3) and the slope of the diagonal as -1/2, we can solve for Vertex 2.

   x = (4 + x2) / 2
   y = (-3 + y2) / 2
   

Solving these equations will give us the coordinates of Vertex 2.

3. Vertex 3: To find this vertex, we can use the equation of the line y = 2x - 1. Since the diagonal passes through this vertex, we can substitute the x-coordinate of Vertex 2 into the equation to find the y-coordinate.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Equations of the Sides

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y2) / 2 ``` Solving these equations will give us the coordinates of Vertex 2.

Vertex 3: Substituting the x-coordinate of Vertex 2 into the equation of the line y = 2x - 1, we have: ``` y = 2x - 1 ``` Substituting the x-coordinate of Vertex 2 into this equation will give us the y-coordinate of Vertex 3.

Now that we have the coordinates of all four vertices, we can find the equations of the sides of the square using the point-slope form of a line.

Let's calculate the coordinates of Vertex 2 and Vertex 3:

Vertex 2: Using the midpoint formula, we have: ``` x = (4 + x2) / 2 -3 = (-3 + y