This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.

This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.

This question tests verifying collinearity via constant slope, with any pairs giving same m, using similar triangles to explain, related to equation derivation. Slope m=(y₂-y₁)/(x₂-x₁) constant for collinear points; similar triangles share angles, proportions like 6/2=12/4=3. For (1,4),(3,8),(7,16), m=(8-4)/(3-1)=4/2=2 and (16-8)/(7-3)=8/4=2. This supports the claim with matching slopes. Errors: inversion to 2 or misstating 4/2=4. Process: (1) calculate m for segments, (2) confirm equality, (3) for equation, check origin or find b, (4) form y=mx+b if needed. Similar triangles prove via proportionality; mistakes include using overall slope alone or inversion.

This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, for (0,1), (2,5), (4,9), slopes (5-1)/(2-0)=4/2=2 and (9-5)/(4-2)=4/2=2, and using (0,1) b=1 so y=2x+1. This matches choice A. Errors include inverting to 1/2 in B, wrong first slope 4 in C, and wrong b=-1 in D. To verify and derive, (1) pick two points for m, (2) verify with another pair, (3) not through origin so solve for b using a point, (4) write y=mx+b. Similar triangles prove constancy as they share angles with parallel sides, making sides proportional and rise/run ratios equal; common mistakes include inverting rise/run, claiming slope varies, or wrong sign for b.

This question tests verifying constant slope on a line, where any two points give the same m, using similar triangles for explanation, and linking to deriving y=mx or y=mx+b equations. Slope m = (y₂ - y₁)/(x₂ - x₁) remains constant, defining lines; similar triangles prove this with equal angles from parallel sides and proportional sides, e.g., 6/2 = 12/4 = 3, showing constant slope; derive y=mx for origin via m=y/x, or y=mx+b with m from points and b from (0,b). For points (2,5),(4,9),(6,13), m between first two: (9-5)/(4-2)=4/2=2, between last two: (13-9)/(6-4)=4/2=2. This confirms constant slope as in choice A. Common errors: inverting to 1/2 in B, stating 4 without dividing in C, or miscalculating second as 4 in D. Steps: (1) pick 2 points, calculate m, (2) verify with another pair, (3) if origin y=mx, else find b via intercept or solving with point, (4) equation. Similar triangles: shared angle, proportional sides from parallelism, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form for origin/non-origin.

This question tests slope constancy for collinearity, where any two points must give same $m$, using similar triangles to explain, and relating to equation derivation like $y=mx$ or $y=mx+b$. Slope $m=(y_2-y_1)/(x_2-x_1)$ is constant on straight lines; similar triangles on the line have equal angles and proportional sides, such as $6/2=12/4=3$, proving uniformity. For $(2,1),(6,9),(8,13)$, $m=(9-1)/(6-2)=8/4=2$ and $(13-9)/(8-6)=4/2=2$. This verifies collinearity with constant slope 2. Errors: inverting to $1/2$ or miscalculating like $8/4=4$. Process: (1) compute $m$ between pairs, (2) ensure they match, (3) if origin $y=mx$, else find $b$ by substitution, (4) write equation. Similar triangles confirm via shared angles and proportions; common mistakes are inversion or assuming overall slope suffices without pair checks.

This question tests that slope is constant on a line, verifiable with any two points giving the same m, explained via similar triangles, and deriving y=mx for origin-passing lines or y=mx+b otherwise. Slope m=(y₂-y₁)/(x₂-x₁) stays constant, a key line property; similar triangles with different sizes on the line share angles (parallel sides) and have proportional sides, e.g., ratios 6/2=12/4=3 remain equal. For points (0,0) and (4,12), m=(12-0)/(4-0)=12/4=3. Since it passes through the origin, the equation is y=3x. Mistakes include calculating slope as 1/3 by inverting or choosing y=mx+b form unnecessarily. Steps: (1) compute m from points, (2) confirm with another pair if available, (3) for origin lines use y=mx, else find b from intercept or point substitution, (4) form equation. Similar triangles prove constancy through proportional rise/run from shared angles; errors like claiming varying slopes or wrong forms for origin lines are common.

This question tests constant slope comparison between lines, each with any points giving same m, explained via similar triangles, and deriving equations like y=mx or y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) is constant per line; similar triangles share angles and proportions, e.g., 6/2=12/4=3. For Line 1 (0,0)(3,9), m=9/3=3; Line 2 (0,2)(3,8), m=6/3=2. Line 1 has steeper slope 3 > 2. Errors: inverting to 1/3 and 1/2 or swapping slopes. Steps: (1) calculate m per line, (2) verify constancy if needed, (3) use y=mx for origin or find b, (4) compare. Similar triangles prove constancy; mistakes include claiming same slope from shared x-change.

This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, the triangles from (0,0) to (2,6) and to (4,12) are similar because they share the angle at origin and right angles, with ratios 6/2=3 and 12/4=3 matching. This explains constant slope, as in choice A. Errors include denying similarity due to size in B, inverting to 1/3 in C, and claiming slope increases in D. To use similar triangles, note they share angles (parallel sides), sides proportional, so rise/run constant; process: calculate ratios, confirm equality. Common mistakes: inverting rise/run, claiming non-similar due to size, using wrong form.

This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, for (0,0), (2,8), (5,20), slopes 8/2=4 and 20/5=4, through origin so y=4x. This matches choice A, explaining constancy and origin. Errors: adding b=8 in B, inverting to 1/4 in C, wrong m=5 in D. To derive, (1) calculate m with pairs, (2) verify equality, (3) through origin so y=mx, (4) write equation. Similar triangles prove constant ratio; mistakes: adding b when origin, inverting, wrong m based on x-value.