Transformations

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Answer Key & Explanations

Transformations · Grade 8 · Worksheet 1

1. Triangle ABC with vertices A(5, 10), B(10, 10), C(5, 15) is reflected over the x-axis, then dilated by a scale factor of 2 from the origin. Find the coordinates of the final image of point C. Answer: (10, -30) Solution: A reflection over the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. A dilation from the origin multiplies both coordinates by the scale factor.
   Full step-by-step solution

   A reflection over the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. A dilation from the origin multiplies both coordinates by the scale factor. When performing multiple transformations, apply them in the given order.

2. Triangle Hana with vertices H(2,4), A(6,4), N(4,8) is reflected over the x-axis. What are the coordinates of the image? Answer: H'(2,-4), A'(6,-4), N'(4,-8) Solution: Reflection over the x-axis is a transformation that flips a figure across the x-axis. This transformation preserves congruence but changes the orientation.
   Full step-by-step solution

   Reflection over the x-axis is a transformation that flips a figure across the x-axis. This transformation preserves congruence but changes the orientation. The rule for reflection over the x-axis is (x,y) → (x,-y), meaning each point's x-coordinate remains unchanged while its y-coordinate becomes the opposite sign.

3. Triangle Emma with vertices E(3,5), M(7,9), M(5,11) is reflected over the line y = x, then rotated 90° counterclockwise about the origin. What are the coordinates of the final image of vertex E? Answer: (-5,-3) Solution: When performing composite transformations, you apply them in order from right to left. The reflection over y=x swaps coordinates, and the rotation changes them according to a specific rule.
   Full step-by-step solution

   When performing composite transformations, you apply them in order from right to left. The reflection over y=x swaps coordinates, and the rotation changes them according to a specific rule. Practice with a simpler point like (1,2) to understand each transformation before applying to the given coordinates.

4. Matiu is designing a triangular garden plot with vertices at coordinates A(2, 4), B(6, 4), and C(4, 8). He creates a similar plot by applying a dilation with a scale factor of 2 and center at the origin. What is the y-coordinate of point C' in the dilated garden plot? Answer: 16 Solution: The original coordinates of point C are (4, 8) The dilation has a scale factor of 2 with center at the origin To find the coordinates after dilation, multiply each coordinate by the scale factor The y-coordinate of C' is 8 × 2 = 16 Therefore, the y-coordinate of point C' is 16
   Full step-by-step solution

   Step 1: The original coordinates of point C are (4, 8) Step 2: The dilation has a scale factor of 2 with center at the origin Step 3: To find the coordinates after dilation, multiply each coordinate by the scale factor Step 4: The y-coordinate of C' is 8 × 2 = 16 Step 5: Therefore, the y-coordinate of point C' is 16

5. Triangle Charlotte with vertices C(8,12), H(11,12), A(8,15) is reflected over the x-axis. What are the coordinates of the image? Answer: C(8,-12), H(11,-12), A(8,-15) Solution: Reflection over the x-axis is a transformation that flips a figure across the horizontal axis.
   Full step-by-step solution

   Reflection over the x-axis is a transformation that flips a figure across the horizontal axis. This transformation preserves the x-coordinates but changes the sign of the y-coordinates, creating a mirror image across the x-axis.

6. Isabella is designing a logo for her school robotics team. She creates a triangular emblem with vertices at A(2, 7), B(7, 7), and C(4, 12). To position it correctly on the team banner, she needs to reflect the triangle across the y-axis. What are the coordinates of the reflected triangle's vertices? Answer: E Solution: A reflection across the y-axis creates a mirror image where each point's x-coordinate changes sign while its y-coordinate remains unchanged.
   Full step-by-step solution

   A reflection across the y-axis creates a mirror image where each point's x-coordinate changes sign while its y-coordinate remains unchanged. This transformation preserves the shape and size of the figure, making the original and reflected triangles congruent.

7. Mere has a triangle with vertices at (2, 4), (6, 4), and (4, 8). She performs two transformations: first a reflection across the x-axis, then a dilation with a scale factor of 2 centered at the origin. How many times larger is the area of the final triangle compared to the original triangle? Answer: 4 Solution: Original triangle has vertices at (2, 4), (6, 4), and (4, 8). After reflection across the x-axis, the vertices become (2, -4), (6, -4), and (4, -8). Reflection preserves area, so area remains the same.
   Full step-by-step solution

   Step 1: Original triangle has vertices at (2, 4), (6, 4), and (4, 8). Step 2: After reflection across the x-axis, the vertices become (2, -4), (6, -4), and (4, -8). Reflection preserves area, so area remains the same. Step 3: After dilation with scale factor 2 centered at the origin, each coordinate is multiplied by 2. The vertices become (4, -8), (12, -8), and (8, -16). Step 4: Dilation with scale factor k increases area by a factor of k². With k = 2, area increases by 2² = 4. Step 5: Therefore, the final triangle's area is 4 times larger than the original triangle's area. The answer is 4.

8. Emma draws a triangle with vertices at (1, 1), (5, 1), and (3, 7). She then applies a dilation centered at the origin with a scale factor of 3. How many times larger is the area of the dilated triangle compared to the original triangle? Answer: 9 Solution: The dilation has a scale factor of 3, meaning all linear dimensions are multiplied by 3. For area calculations, the scale factor is applied to both length and width dimensions.
   Full step-by-step solution

   Step 1: The dilation has a scale factor of 3, meaning all linear dimensions are multiplied by 3. Step 2: For area calculations, the scale factor is applied to both length and width dimensions. Step 3: Area scale factor = (linear scale factor)^2 = 3^2 = 9. Step 4: Therefore, the dilated triangle's area is 9 times larger than the original triangle's area. The answer is 9.