Question

Line g: y=−4/5x+7/5 line h: y=5/4x+3/4 Is line g perpendicular to line h? Why or why not?

A. Yes, because the slopes of lines g and h are opposite and the y-intercepts are different.
B. No, because the slopes of lines g and h have different signs.
C. No, because the y-intercepts of lines g and h are different.
D. Yes, because the slopes of lines g and h are opposite and reciprocal.

Answer 1

Line g with a slope of -4/5 and line h with a slope of 5/4 are perpendicular because their slopes are negative reciprocals of each other, their product is -1. Hence, the correct answer is D. Yes, because the slopes of lines g and h are opposite and reciprocal.

Step-by-step explanation:

To determine if two lines are perpendicular, we need to compare their slopes. Two lines are perpendicular if and only if the product of their slopes is -1, meaning that the slopes are negative reciprocals of each other. For the lines described:

• Line g: y = -4/5x + 7/5 has a slope of -4/5.
• Line h: y = 5/4x + 3/4 has a slope of 5/4.

By multiplying the slopes of line g and line h:

-4/5 × 5/4 = -1

Since the product of the slopes is -1, line g is indeed perpendicular to line h. The correct answer to whether line g is perpendicular to line h is: D. Yes, because the slopes of lines g and h are opposite and reciprocal. Answer option A is incorrect because although line g and h are perpendicular, the reasoning that 'the y-intercepts are different' is not a criterion for perpendicularity. Options B and C are both incorrect as they do not correctly identify the relationship between the slopes of perpendicular lines.