The gradients of several lines are as follows:
Line a b c d e f g h
Gradient −3 0.5 0.4
(a) Find two pairs of lines that are parallel to each other.
(b) Find any two pairs of lines that are at right angles to each other.
(a) Two pairs of lines that are parallel to each other are:
Lines a and g both have a gradient of -3.
Lines b and c both have a gradient of 0.5.
(b) Two pairs of lines that are at right angles to each other are:
Lines a and f have gradients of -3 and 0.4, respectively. The product of their gradients is -3 * 0.4 = -1.2, which means they are perpendicular to each other.
Lines c and h have gradients of 0.5 and 0, respectively. The product of their gradients is 0.5 * 0 = 0, which means they are perpendicular to each other.
(a) To find two pairs of lines that are parallel to each other, we need to compare their gradients.
Pair 1: Line a and Line c
Since the gradient of Line a is -3 and the gradient of Line c is also -3, Line a and Line c are parallel.
Pair 2: Line e and Line f
Since the gradient of Line e is 0.5 and the gradient of Line f is also 0.5, Line e and Line f are parallel.
(b) To find any two pairs of lines that are at right angles to each other, we need to determine if the product of their gradients is -1.
Pair 1: Line a and Line h
The gradient of Line a is -3. We need to find a line with a gradient that is the negative reciprocal of -3. The negative reciprocal of -3 is 1/3. Since the gradient of Line h is 1/3, Line a and Line h are at right angles to each other.
Pair 2: Line c and Line g
The gradient of Line c is 0.4. We need to find a line with a gradient that is the negative reciprocal of 0.4. The negative reciprocal of 0.4 is -2.5. Since the gradient of Line g is -2.5, Line c and Line g are at right angles to each other.