Question

Доведіть, що чотирикутник ABCD є прямокутником, якщо А(-5; 1), B(-1; -1), C(-2;-3) D(-6; -1). Люди очень срочно плиз даю 100 балов !!!!!!!!!!!​

Answer 1

Например так:

Сдвинем всю фигуру так, что А в начале координат

Получим А1,В1,С1,Д1

А1(0,0),В1(4,-2),С1(3,-4),Д1(-1,-2)

Угол Д1А1В1- равен 90 градусов,т.к. скалярное произведение векторов  В1 и Д1 рано 0.

Сдвинем точку В в начало координат

Получим А2В2С2Д2

А2(4,-2),В2(0,0),С2(-1,-2),Д2(-5,0)

УГОЛ  А2В2С2=90 градусов

А2*С2=0   (4*-1)+(-2*-2)=0

Сдвинем точку  С в начало координат

А3(-3,4),В3(1,2),С3(0,0),Д3(-4,2)

угол В3С3Д3  90 градусов

В3*Д3=0  1*(-4)+2*2=0

Три угла по 90 градусов, четвертый можно не проверять, а можно и проверить также.

Answer 2

I can help you with your geometry question. To prove that the quadrilateral ABCD is a rectangle, you need to show that it has four right angles and opposite sides are parallel and congruent. One way to do this is to use the slope formula and the distance formula to calculate the slopes and lengths of the sides. Then, you can use the following facts:

- If two lines have the same slope, they are parallel. - If two lines have slopes that are negative reciprocals of each other, they are perpendicular. - If two lines are perpendicular, they form a right angle. - If a quadrilateral has four right angles, it is a rectangle.

Using the slope formula, we get:

- Slope of AB = (y2 - y1) / (x2 - x1) = (-1 - 1) / (-1 - (-5)) = -2 / 4 = -1 / 2 - Slope of BC = (-3 - (-1)) / (-2 - (-1)) = -2 / -1 = 2 - Slope of CD = (-1 - (-3)) / (-6 - (-2)) = 2 / -4 = -1 / 2 - Slope of DA = (1 - (-1)) / (-5 - (-6)) = 2 / 1 = 2

Using the distance formula, we get:

- Length of AB = √((x2 - x1)^2 + (y2 - y1)^2) = √((-1 - (-5))^2 + (-1 - 1)^2) = √(16 + 4) = √20 - Length of BC = √((-2 - (-1))^2 + (-3 - (-1))^2) = √(1 + 4) = √5 - Length of CD = √((-6 - (-2))^2 + (-1 - (-3))^2) = √(16 + 4) = √20 - Length of DA = √((-5 - (-6))^2 + (1 - (-1))^2) = √(1 + 4) = √5

Now, we can see that:

- AB ∥ CD and BC ∥ DA, because they have the same slope. - AB ⊥ BC and CD ⊥ DA, because their slopes are negative reciprocals of each other. - AB = CD and BC = DA, because they have the same length.

Therefore, ABCD is a rectangle by definition. For more information, you can check out these web pages: [How to Prove that a Quadrilateral Is a Rectangle - dummies](https://www.dummies.com/article/academics-the-arts/math/geometry/how-to-prove-that-a-quadrilateral-is-a-rectangle-188109/), [Prove it is a Rectangle - Softschools.com](https://www.softschools.com/math/geometry/proofs/prove_it_is_a_rectangle/), [How do you prove a quadrilateral is a rectangle with coordinates?](https://profound-tips.com/blog/how-do-you-prove-a-quadrilateral-is-a-rectangle-with-coordinates/), [Q: How can one prove that a quadrilateral is a rectangle?](https://www.ck12.org/flexi/geometry/quadrilaterals/how-can-one-prove-that-a-quadrilateral-is-a-rectangle/). I hope this helps. Have a nice day!