Question

РЕШИТЕ СРОЧНО! Даны точки А(-3,3) В(-1,6) С(2,4) Д(0,1). Докажите, что АВСD -ромб.

Answer 1

Ответ:

Чтобы доказать, что АВСD -ромб, надо чтобы растояние между А и В

было таким же как ростояние между С и Д.

Растояние между точками находится по формуле

АВ=√(х2-х1)²+(у2-у1)²

АВ=√(-1+3)²+(6-3)²=√13

СД=√(0-2)²+(1-4)²=√13

АВ=СД. Доказано

Answer 2

Proof that ABCD is a Rhombus

To prove that ABCD is a rhombus, we need to show that all four sides of the quadrilateral are equal in length and that the opposite sides are parallel to each other.

Let's start by calculating the lengths of the sides:

- Side AB: The distance between points A(-3,3) and B(-1,6) can be calculated using the distance formula:

%5E2%20%2B%20(y_2%20-%20y_1)%5E2%7D)

Substituting the coordinates, we get:

)%5E2%20%2B%20(6%20-%203)%5E2%7D%20%3D%20%5Csqrt%7B2%20%2B%209%7D%20%3D%20%5Csqrt%7B11%7D)

- Side BC: The distance between points B(-1,6) and C(2,4) can be calculated using the distance formula:

)%5E2%20%2B%20(4%20-%206)%5E2%7D%20%3D%20%5Csqrt%7B9%20%2B%204%7D%20%3D%20%5Csqrt%7B13%7D)

- Side CD: The distance between points C(2,4) and D(0,1) can be calculated using the distance formula:

%5E2%20%2B%20(1%20-%204)%5E2%7D%20%3D%20%5Csqrt%7B4%20%2B%209%7D%20%3D%20%5Csqrt%7B13%7D)

- Side DA: The distance between points D(0,1) and A(-3,3) can be calculated using the distance formula:

%5E2%20%2B%20(3%20-%201)%5E2%7D%20%3D%20%5Csqrt%7B9%20%2B%204%7D%20%3D%20%5Csqrt%7B13%7D)

Now, let's compare the lengths of the sides:

- AB = CD =  - BC = DA = 

Since all four sides have equal lengths, we have shown that AB = BC = CD = DA.

Next, let's prove that the opposite sides are parallel to each other. We can calculate the slopes of the sides to determine if they are parallel:

- Slope of AB: The slope between points A(-3,3) and B(-1,6) can be calculated using the slope formula:

%7D%20%3D%20%5Cfrac%7B3%7D%7B2%7D)

- Slope of BC: The slope between points B(-1,6) and C(2,4) can be calculated using the slope formula:

%7D%20%3D%20%5Cfrac%7B-2%7D%7B3%7D)

- Slope of CD: The slope between points C(2,4) and D(0,1) can be calculated using the slope formula:


- Slope of DA: The slope between points D(0,1) and A(-3,3) can be calculated using the slope formula:


The slopes of AB and CD are both , and the slopes of BC and DA are both . Since the slopes of the opposite sides are equal, we can conclude that AB is parallel to CD and BC is parallel to DA.

Therefore, we have shown that all four sides of ABCD are equal in length and that the opposite sides are parallel to each other, which proves that ABCD is a rhombus.

Note: The proof provided above is based on the given coordinates of the points A, B, C, and D.