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ella [17]
2 years ago
14

Given A (3,4), B (4,10), C (-9,1), D (-7,13), determine whether lines AB and CD are parallel, perpendicular or neither

Mathematics
1 answer:
yuradex [85]2 years ago
8 0

Answer:

parallel

Step-by-step explanation:

A (3,4), B (4,10)

Slope of AB = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\   =\frac{10-4}{4-3}\\\\   =\frac{6}{1}\\\\   = 6

C (-9,1), D (-7,13)

Slope of CD = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\   =\frac{13-1}{-7-[-9]}\\\\   =\frac{12}{-7+9}\\\\   = \frac{12}{2}\\\\   = 6

Slope of AB = Slope CD = 6

If two lines have same slope, they are parallel lines

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For A (1, –1), B (–1, 3), and C (4, –1), find a possible location of a fourth point, D, so that a parallelogram is formed using
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The possible coordinates for the vertex D of the parallelogram could be (-4, 3)

It is given that A(1, –1), B (–1, 3), and C (4, –1) are the three vertices of the parallelogram.

Let us assume that the vertices are in the order A, C, B, and D where the coordinates of D are (a, b).

In this scenario, if we join AB and CD, they will become the diagonals of the parallelogram ABCD.

According to the properties of a parallelogram, diagonals bisect each other.

Hence, mid-point of AB = mid-point of CD

Now, according to the mid-point theorem, if mid-point of AB is given as (x,y), then,

x = (x₁ + x₂)/2 and y = (y₁ + y₂)/2

Here, for AC,

x₁ = 1, y₁ = -1

x₂ = -1, y₂ = 3

Then, x = ( 1 - 1)/2 and y = (-1 + 3)/2

(x, y) ≡ (0, 1)  ............. (1)

Since (x, y) is also the mid-point of CD, we also have,

x₁ = 4, y₁ = -1

x₂ = a, y₂ = b

Then, x = (4 + a)/2 and y = (-1 + b)/2

(x, y) ≡ ((4 + a)/2, (-1 + b)/2) ................... (2)

From (1) and (2),

(4+a)/2 = 0 and (-1+b)/2 = 1

4+a = 0 and (-1+b) = 2

a = -4 and b = 3

Hence, the fourth vertex D of the parallelogram can be possibly located at (-4, 3)

Learn more about a parallelogram here:

brainly.com/question/1563728

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