Final answer:
To find the distance between points P and Q, we use the slopes provided to find their coordinates and then apply the distance formula. Points P and Q are found to be (16, 12) and (-18, 12) respectively, and the distance between them is 34 units.
Step-by-step explanation:
The student is asking to determine the distance between two points, P and Q, which lie on different lines with given slopes, and these lines intersect at the origin. We can find the coordinates of points P and Q by using the information that each line passes through the origin (which gives us the y-intercept) and by applying the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. For the line containing point P(16,p) with a slope of 3/4, the line's equation is y = (3/4)x because it passes through the origin (0,0). Thus, p can be found by substituting x = 16 into the equation: p = (3/4) × 16 = 12. So, point P becomes (16, 12).
For the line containing point Q(-18,q) with a slope of -2/3, the line's equation is y = (-2/3)x. Finding q involves substituting x = -18: q = (-2/3) × -18 = 12. Hence, point Q is (-18, 12). To find the distance between points P and Q, use the distance formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. Substituting the coordinates of P and Q into the formula, we get d = \sqrt{(16 - (-18))^2 + (12 - 12)^2} = \sqrt{(34)^2} = 34 units. Therefore, the distance between points P and Q is 34 units.