Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?(1 point)

Responses

No…these lines are parallel
No…these lines are parallel

Yes…these lines are perpendicular
Yes…these lines are perpendicular

You cannot tell without a graph
You cannot tell without a graph

No…these lines are perpendicular
No…these lines are perpendicular

Yes…these lines are not parallel
Yes…these lines are not parallel

Yes…these lines are parallel

1 answer

To determine if the two lines intersect, we first need to calculate the slopes of both lines.

Line 1 passes through the points (3, 7) and (-2, -5). The slope (m₁) is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For (3, 7) and (-2, -5):

  • \( x_1 = 3, y_1 = 7 \)
  • \( x_2 = -2, y_2 = -5 \)

Calculating the slope:

\[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]

Line 2 passes through the points (4, 8) and (10, -2). Calculating the slope (m₂) as follows:

For (4, 8) and (10, -2):

  • \( x_1 = 4, y_1 = 8 \)
  • \( x_2 = 10, y_2 = -2 \)

Calculating the slope:

\[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]

Now we have the slopes:

  • Slope of Line 1 (m₁) = \( \frac{12}{5} \)
  • Slope of Line 2 (m₂) = \( -\frac{5}{3} \)

Since the slopes are not equal, the lines are not parallel.

To check for perpendicularity, we need to see if the product of the slopes is -1:

\[ m_1 \times m_2 = \left(\frac{12}{5}\right) \times \left(-\frac{5}{3}\right) = -\frac{12}{3} = -4 \]

Since the product of the slopes is not -1, the lines are not perpendicular.

Therefore, the lines intersect and they are not parallel.

The correct answer is: Yes…these lines are not parallel.