Question

Даны точки A(1;0); B(x;8); M(7;1) и N(x;0). Найди значение x и напиши координаты B и N, если расстояние между точками A и B такое же, как между точками M и N. (Если это необходимо, округли результат до тысячных.) B(;8); N(;0).

Answer 1

Ответ:

В(7,5; 2)

N(7,5; 0)

Объяснение:

Расстояние между точками АВ и MN вычисляется по формуле:

АВ²=(Хв–Ха)²+(Ув–Уа)²;

MN²=(Xn–Xm)²+(Уn–Уm)²

и так как эти расстояния равны, составим уравнение:

(Хв–Ха)²+(Ув–Уа)²=(Xn–Xm)²(Уn–Уm)²

(х–4)²+(2–0)²=(х–8)²+(0–4)²

х²–8х+16+2²=х²–16х+64+(–4)²

х²–8х+16+4=х²–16х+64+16

х²–8х+20=х²–16х+80

х²–8х–х²+16х=80–20

8х=60

х=60÷8

х=7,5

Answer 2

Given Information:

We are given the following points: - A(1,0) - B(x,8) - M(7,1) - N(x,0)

We need to find the value of x and the coordinates of points B and N, such that the distance between points A and B is the same as the distance between points M and N.

Solution:

To find the value of x and the coordinates of points B and N, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can set up two equations:

1. For the distance between points A and B: sqrt((x - 1)^2 + (8 - 0)^2) = sqrt((7 - x)^2 + (1 - 0)^2)

2. For the distance between points M and N: sqrt((x - 7)^2 + (0 - 1)^2) = sqrt((x - x)^2 + (0 - 0)^2)

Simplifying these equations, we get:

1. (x - 1)^2 + 64 = (7 - x)^2 + 1 2. (x - 7)^2 + 1 = 0

Solving these equations will give us the value of x and the coordinates of points B and N.

Let's solve these equations step by step:

1. (x - 1)^2 + 64 = (7 - x)^2 + 1

Expanding the squares: x^2 - 2x + 1 + 64 = x^2 - 14x + 49 + 1

Simplifying: x^2 - 2x + 65 = x^2 - 14x + 50

Subtracting x^2 from both sides: -2x + 65 = -14x + 50

Adding 14x to both sides: 12x + 65 = 50

Subtracting 65 from both sides: 12x = -15

Dividing by 12: x = -15/12 = -1.25

So, the value of x is -1.25.

Now, let's find the coordinates of points B and N using the value of x:

For point B: - x = -1.25 - y = 8

So, the coordinates of point B are (-1.25, 8).

For point N: - x = -1.25 - y = 0

So, the coordinates of point N are (-1.25, 0).

Therefore, the value of x is -1.25, and the coordinates of points B and N are (-1.25, 8) and (-1.25, 0) respectively.

Note: The coordinates of point A and the distance between points A and B are not required to find the solution.

Answer:

The value of x is -1.25, and the coordinates of points B and N are (-1.25, 8) and (-1.25, 0) respectively.