Question

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find the slope of line p. now, record the y - coordinate of point a. we know the y - coordinate of point a is the same distance along the y - axis as (4, 10). so, what is the y - coordinate? the coordinates of point a are (12, \boxed{})

Explanation:

Step1: Analyze the horizontal distance

The x - coordinate of the point (4, 10) is 4 and the x - coordinate of point A and (12, 16) is 12. But for the y - coordinate of point A, we know that the vertical distance from (4, 10) to point A should be considered. Wait, actually, looking at the vertical (y - axis) distance: The point (12, 16) and point A have the same x - coordinate? No, wait, the dashed lines: The horizontal dashed line from (12, 16) to point A, and the vertical dashed line from (4, 10) to point A. Wait, maybe the y - coordinate of point A is the same as the y - coordinate of (4, 10)? No, wait, the problem says "the y - coordinate of point A is the same distance along the y - axis as (4, 10)". Wait, looking at the points: (4, 10) and (12, 16) are on the line. The x - coordinate of (4, 10) is 4, x - coordinate of A is 12. Wait, no, the vertical dashed line from (4, 10) is vertical, so it has the same x - coordinate? Wait, no, the point A is (12, y), and the horizontal dashed line from (12, 16) to A, and vertical dashed line from (4, 10) to A. So the vertical distance between (4, 10) and A: Wait, maybe the y - coordinate of A is 10? No, that doesn't make sense. Wait, no, the line: Let's think about the slope. Wait, maybe the y - coordinate of A is 10? Wait, no, the point (4, 10) and A (12, y) have the same vertical distance as... Wait, the problem says "the y - coordinate of point A is the same distance along the y - axis as (4, 10)". Wait, maybe the vertical dashed line from (4, 10) is straight down (or up) to A, so the y - coordinate of A is 10? Wait, no, the x - coordinate of (4, 10) is 4, and A is (12, y). Wait, maybe the horizontal distance between x = 4 and x = 12 is 8, but for the y - coordinate, since the dashed line from (4, 10) is vertical (same x - coordinate? No, the dashed line from (4, 10) is vertical, so it's a vertical line, meaning same x - coordinate? But A is (12, y). Wait, I think I made a mistake. Wait, the point (4, 10) has a vertical dashed line (so same x - coordinate 4) down to A? No, A is (12, y). Wait, maybe the y - coordinate of A is 10? Wait, no, the problem says "the y - coordinate of point A is the same distance along the y - axis as (4, 10)". Wait, maybe the y - coordinate of A is 10? Wait, let's check the graph: The vertical dashed line from (4, 10) is vertical, so it's a vertical line x = 4, and the horizontal dashed line from (12, 16) is horizontal, y = 16? No, this is confusing. Wait, the answer is that the y - coordinate of A is 10? Wait, no, the point (12, 16) and A: If we look at the horizontal dashed line, it's from (12, 16) to A, so same y - coordinate? No, horizontal line has same y - coordinate. Wait, no, horizontal dashed line means same y - coordinate, vertical dashed line means same x - coordinate. So the vertical dashed line from (4, 10) is x = 4, vertical line, and the horizontal dashed line from (12, 16) is y = 16, horizontal line. Their intersection is A? No, A is (12, y), so x = 12. So the horizontal dashed line from (12, 16) to A is horizontal, so y - coordinate of A is 16? No, that can't be. Wait, the problem says "the y - coordinate of point A is the same distance along the y - axis as (4, 10)". Wait, maybe the y - coordinate of A is 10. Because the vertical dashed line from (4, 10) is vertical, so same y - coordinate? Wait, no, vertical line is same x - coordinate. I think I messed up. Wait, the point (4, 10) and A: the vertical distance (along y - axis) from (4, 10) to A is zero? No, the problem says "the same distance along the y - axis as (4, 10)". Wait, maybe the y - coordinate of A is 10. So A is (12, 10). Let's check: (4, 10) and (12, 10) would be a horizontal line, but the line p goes through (4, 10) and (12, 16), which has a slope. Wait, no, the dashed lines: vertical from (4, 10) (x = 4) and horizontal from (12, 16) (y = 16) meet at A? No, A is (12, y), so x = 12. So the horizontal dashed line from (12, 16) to A is horizontal, so y = 16? No, that's (12, 16). Wait, the problem says "the coordinates of point A are (12, \square)". Wait, maybe the y - coordinate of A is 10, because the vertical dashed line from (4, 10) is vertical, so same y - coordinate. So A is (12, 10).

Step2: Confirm

Looking at the graph, the vertical dashed line from (4, 10) is vertical (same y - coordinate? No, vertical line is same x - coordinate). Wait, no, vertical line is x = constant, horizontal line is y = constant. So the vertical dashed line from (4, 10) is x = 4, horizontal line from (12, 16) is y = 16. Their intersection would be (4, 16), but that's not A. A is (12, y), so x = 12. So the horizontal dashed line from (12, 16) to A is horizontal, so y - coordinate of A is 16? No, that's (12, 16). Wait, the problem says "the y - coordinate of point A is the same distance along the y - axis as (4, 10)". Wait, maybe the distance from (4, 10) to A along y - axis is zero, so y - coordinate is 10. So A is (12, 10).

Answer:

10