Question

Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?(1 point)Responses Yes…these lines are parallelYes…these lines are parallelNo…these lines are parallelNo…these lines are parallelNo…these lines are perpendicularNo…these lines are perpendicular Yes…these lines are perpendicular Yes…these lines are perpendicular You cannot tell without a graphYou cannot tell without a graphYes…these lines are not parallel

Answer 1

To determine whether the two lines intersect, we can start by calculating the slopes of both lines.

1. Slope of the first line through (3, 7) and (-2, -5): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]

2. Slope of the second line through (4, 8) and (10, -2): \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]

Now that we have the slopes:

• First line slope: \( \frac{12}{5} \)
• Second line slope: \( -\frac{5}{3} \)

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

Next, we can check if the lines are perpendicular. Two lines are perpendicular if the product of their slopes equals -1.

Calculating the product of the slopes: \[ \frac{12}{5} \times -\frac{5}{3} = -\frac{60}{15} = -4 \]

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

Therefore, the two lines are neither parallel nor perpendicular, and they will intersect.

The response should be: Yes…these lines are not parallel.