Both sequences result in the same final point $$(-10, -6)$$. Rotation and dilation about the origin commute because both preserve ratios of distances from the origin. Choice B suggests reflection symmetry that doesn't exist. Choice C incorrectly suggests different distances from origin. Choice D is partially correct about equal distances but wrong about the positions being different.

When you see a dilation problem, you're working with transformations that resize figures while keeping them centered at a specific point. The key is finding the scale factor by comparing corresponding coordinates.

To find the scale factor $$k$$, compare the original point $$M(6, -8)$$ with its image $$M'(9, -12)$$. Since dilation multiplies each coordinate by the scale factor, you have:

• For the x-coordinate: $$6k = 9$$, so $$k = \frac{9}{6} = \frac{3}{2}$$
• For the y-coordinate: $$-8k = -12$$, so $$k = \frac{-12}{-8} = \frac{3}{2}$$

Both coordinates give the same scale factor of $$\frac{3}{2}$$, confirming our calculation. Now apply this scale factor to point $$N(-4, 10)$$:

• New x-coordinate: $$-4 \times \frac{3}{2} = -6$$
• New y-coordinate: $$10 \times \frac{3}{2} = 15$$

So $$N'(-6, 15)$$, which is answer choice C.

Looking at the wrong answers: A) $$(-2, 12)$$ suggests using scale factor $$\frac{1}{2}$$ for x and $$\frac{6}{5}$$ for y, mixing up the calculation. B) $$(-6, 12)$$ correctly finds the x-coordinate but incorrectly uses scale factor $$\frac{6}{5}$$ for the y-coordinate. D) $$(-8, 15)$$ gets the y-coordinate right but doubles the x-coordinate instead of applying the scale factor.

Remember: in dilation problems, always verify your scale factor works for both coordinates of the given transformation before applying it to the new point. This prevents calculation errors and ensures consistency.

When you encounter coordinate transformation problems, your first step is to analyze how each point changes to identify the pattern. Look at what happens to the x and y coordinates systematically.

Let's examine the given transformations: $$P(3,1) \to P'(1,3)$$, $$Q(7,1) \to Q'(1,7)$$, and $$R(9,5) \to R'(5,9)$$. Notice that in each case, the x-coordinate and y-coordinate are swapped. This is the defining characteristic of a reflection across the line $$y = x$$.

To find $$S'$$, we first need the coordinates of point $$S$$. Since $$PQRS$$ is a parallelogram, opposite sides are parallel and equal. Using the given points, we can determine that $$S(5,5)$$. When reflected across $$y = x$$, point $$S(5,5)$$ becomes $$S'(5,5)$$ because swapping coordinates that are equal gives the same point.

Now let's examine why the other answers are incorrect. Choice A suggests a $$90°$$ counterclockwise rotation about the origin, which would map $$(x,y)$$ to $$(-y,x)$$. This would send $$P(3,1)$$ to $$P'(-1,3)$$, not $$(1,3)$$. Choice C has the correct transformation but gives $$S'(1,5)$$ instead of $$(5,5)$$ – this reflects a calculation error in finding point $$S$$. Choice D mentions a translation followed by reflection, but the pattern shows a simple single transformation.

Study tip: When identifying transformations, always check what happens to coordinates systematically. Reflection across $$y = x$$ always swaps x and y coordinates, making it easy to spot this transformation type.

Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, the translation by (4,3) uses the rule (x,y)→(x+4,y+3), which matches choice C. A common error is sign wrong in translation, like writing (x+4,y+3) as (x-4,y-3) or mixing the values like (x+3,y+4). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point B(2,1) reflected over y=x: apply (x,y)→(y,x) getting B'(1,2), or over y-axis would be (-2,1). In this case, the reflection over y=x correctly applies (x,y)→(y,x) to transform B(2,1) to B'(1,2). A common error might be using the wrong reflection rule, such as (-x,y) instead of (y,x) for y=x, or swapping incorrectly to (2,-1). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, the 180° rotation uses the rule (x,y)→(-x,-y), which matches choice B. A common error is rotation formula wrong, like confusing 180° with 90° as (-y,x) instead of (-x,-y). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

Tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky). Apply rule to all vertices getting image figure. For example, triangle A(1,2),B(3,2),C(2,4) translated by (4,3): apply (x,y)→(x+4,y+3) getting A'(5,5),B'(7,5),C'(6,7), or reflection over y-axis: (x,y)→(-x,y) giving A'(-1,2),B'(-3,2),C'(-2,4). In this case, reflecting C(5,2) over the y-axis gives C'(-5,2) using (x,y)→(-x,y), which matches choice B. A common error is wrong coordinate negated, like negating y instead of x for y-axis reflection, resulting in (5,-2). Applying rule: (1) identify transformation type and parameters, (2) write coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky). Mistakes: sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), forgetting to apply to all coordinates (does x but not y).

This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point R(6,-1) rotated 180°: apply (x,y)→(-x,-y) getting R'(-6,1), or 90° CCW would be (1,6). In this case, the 180° rotation correctly applies (x,y)→(-x,-y) to transform R(6,-1) to R'(-6,1). A common error might be using the wrong rotation formula, such as (-y,x) for 180° instead of (-x,-y), or miscalculating signs to (6,1). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, square W(1,-2) translated to W'(-2,2): the change is -3 in x and +4 in y, so (x-3,y+4), applied to others confirms. In this case, the translation correctly applies the rule (x,y)→(x-3,y+4) to produce W'(-2,2), X'(0,2), Y'(0,4), and Z'(-2,4). A common error might be reversing the signs, such as (x+4,y-3) instead of (x-3,y+4), or miscalculating the vector as (x+3,y-4). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).

This question tests describing transformation effects on coordinates using rules: translation (x,y)→(x+h,y+k), reflection (negate appropriate coordinate), rotation (formula based on angle), dilation (multiply by scale factor). Each transformation has a coordinate rule: translation by (h,k) adds to coordinates (x,y)→(x+h,y+k), reflection over y-axis negates x (x,y)→(-x,y), over x-axis negates y (x,y)→(x,-y), rotation 90° CCW about origin uses (x,y)→(-y,x), dilation scale k from origin multiplies both (x,y)→(kx,ky); apply the rule to all vertices to get the image figure. For example, point Q(-4,5) reflected over the x-axis: apply (x,y)→(x,-y) getting Q'(-4,-5), or over y-axis would be (4,5). In this case, the x-axis reflection correctly applies (x,y)→(x,-y) to transform Q(-4,5) to Q'(-4,-5). A common error might be negating the wrong coordinate, such as using (-x,y) for x-axis instead of (x,-y), or flipping signs incorrectly to (4,5). To apply the rule: (1) identify the transformation type and parameters, (2) write the coordinate rule ((x,y)→...), (3) apply to each vertex (substitute coordinates, calculate image), (4) verify it looks reasonable (translation shifts, reflection flips, rotation turns, dilation resizes). Memorize common rules: translation adds (h,k), x-axis reflection (x,-y), y-axis reflection (-x,y), 90° CCW rotation (-y,x), 180° rotation (-x,-y), dilation scale k is (kx,ky); mistakes include sign errors (most common: wrong sign on translation or reflection), coordinate order (rotation formulas must be exact: (-y,x) not (y,-x)), or forgetting to apply to all coordinates (does x but not y).