Transformation Properties

lessonbunny.com

Answer Key & Explanations

Transformation Properties · Grade 8 · Worksheet 1

1. Emma has a triangular piece of fabric with side lengths of 7 inches, 9 inches, and 11 inches, and angles measuring 41°, 59°, and 80°. She reflects the fabric across a line. Are the side lengths and angle measures preserved after the reflection? Explain which properties of the triangle remain the same and why. Answer: Yes, both side lengths and angle measures are preserved after reflection. Solution: A reflection is a type of rigid transformation (also called an isometry). Rigid transformations preserve distances between points and angle measures. The original triangle has side lengths of 7 inches, 9 inches, and 11 inches.
   Full step-by-step solution

   Step 1: A reflection is a type of rigid transformation (also called an isometry). Rigid transformations preserve distances between points and angle measures. Step 2: The original triangle has side lengths of 7 inches, 9 inches, and 11 inches. After reflection, the image triangle will have exactly the same side lengths because the distance between any two points in the original triangle is unchanged when reflected. Step 3: The original triangle has angles of 41°, 59°, and 80°. After reflection, each angle measure remains the same because reflection does not change the shape or size of angles. Step 4: Therefore, both side lengths (distances) and angle measures are preserved. The reflected triangle is congruent to the original triangle. The answer is: Yes, both side lengths and angle measures are preserved after reflection.

2. Noah draws a triangle with vertices at A(1, 1), B(1, 6), and C(6, 1) on a coordinate grid. He then applies a dilation centered at the origin with a scale factor of 2. After the dilation, he reflects the new triangle across the x-axis. Which properties of the original triangle are preserved through both transformations: the angle measures, the side lengths, and/or the parallelism between the sides? Explain your reasoning. Answer: Only angle measures and parallelism are preserved; side lengths are not preserved. Solution: Identify the original triangle's properties. The triangle has vertices at (1,1), (1,6), and (6,1). It is a right triangle with a right angle at (1,1).
   Full step-by-step solution

   Step 1: Identify the original triangle's properties. The triangle has vertices at (1,1), (1,6), and (6,1). It is a right triangle with a right angle at (1,1). The side lengths are: from (1,1) to (1,6) is 5 units, from (1,1) to (6,1) is 5 units, and from (1,6) to (6,1) is sqrt((6-1)^2 + (1-6)^2) = sqrt(25 + 25) = sqrt(50) = 5*sqrt(2) units. The sides are parallel to the axes (horizontal and vertical lines). Step 2: Apply dilation with scale factor 2 centered at origin. New vertices: A'(2,2), B'(2,12), C'(12,2). Dilation multiplies all distances by 2, so side lengths become 10, 10, and 10*sqrt(2). Angle measures remain unchanged because dilation preserves angles. Parallelism is preserved because lines that were parallel remain parallel after dilation. Step 3: Apply reflection across the x-axis. New vertices: A''(2,-2), B''(2,-12), C''(12,-2). Reflection preserves all distances and angles, so the side lengths remain 10, 10, and 10*sqrt(2). Angle measures stay the same. Parallelism is preserved because parallel lines remain parallel after reflection. Step 4: Conclusion. Through both transformations, angle measures are preserved (dilation and reflection both preserve angles). Parallelism is preserved (both transformations preserve parallelism). Side lengths are NOT preserved because the dilation changed the side lengths from 5, 5, 5*sqrt(2) to 10, 10, 10*sqrt(2). Therefore, only angle measures and parallelism are preserved. The answer is: Only angle measures and parallelism are preserved; side lengths are not preserved.

3. Aroha is designing a triangular park sign. The original triangle has side lengths of 9 cm, 13 cm, and 15 cm, and angles of 35°, 55°, and 90°. She reflects the triangle across a vertical line. Which properties of the triangle are preserved under reflection? Explain your reasoning for angles, side lengths, and whether the shape remains congruent. Answer: Angles, side lengths, and congruence are all preserved; the triangle remains congruent after reflection. Solution: Understand reflection. A reflection is a rigid transformation (isometry) that flips a shape over a line. The original side lengths are 9 cm, 13 cm, and 15 cm.
   Full step-by-step solution

   Step 1: Understand reflection. A reflection is a rigid transformation (isometry) that flips a shape over a line. Rigid transformations preserve distances and angles. Step 2: Check side lengths. The original side lengths are 9 cm, 13 cm, and 15 cm. After reflection, each point moves to an opposite position across the line, but the distance between any two points stays the same. So all three side lengths remain 9 cm, 13 cm, and 15 cm. Distances are preserved. Step 3: Check angles. The original angles are 35°, 55°, and 90°. Reflection does not change the measure of angles; it only changes orientation. So all three angles remain 35°, 55°, and 90°. Angles are preserved. Step 4: Check congruence. Since both side lengths and angles are preserved, the reflected triangle is exactly the same size and shape as the original. The two triangles are congruent. Congruence is preserved. The answer is: Angles, side lengths, and congruence are all preserved; the triangle remains congruent after reflection.

4. A scientist is studying bacteria growth in a lab. The initial population is 500 bacteria, and it doubles every 3 hours. The scientist uses the exponential growth formula P = 500 × 2^(t/3), where P is the population and t is time in hours. After how many hours will the bacteria population reach 8,000? Answer: 12 Solution: Set up the equation: 500 × 2^(t/3) = 8000 Divide both sides by 500: 2^(t/3) = 16 Recognize that 16 is 2^4, so: 2^(t/3) = 2^4 Since the bases are equal, set the exponents equal: t/3 = 4 Multiply both sides by 3: t = 12 Check: 500 × 2^(12/3) = 500 × 2^4 = 500 × 16 = 8000 The answer is 12 hours.
   Full step-by-step solution

   Step 1: Set up the equation: 500 × 2^(t/3) = 8000 Step 2: Divide both sides by 500: 2^(t/3) = 16 Step 3: Recognize that 16 is 2^4, so: 2^(t/3) = 2^4 Step 4: Since the bases are equal, set the exponents equal: t/3 = 4 Step 5: Multiply both sides by 3: t = 12 Step 6: Check: 500 × 2^(12/3) = 500 × 2^4 = 500 × 16 = 8000 The answer is 12 hours.

5. Liam is designing a rectangular garden with a length that is 3 feet more than twice its width. If the area of the garden is 65 square feet, what are the dimensions of the garden? Answer: width = 5 feet, length = 13 feet Solution: Let the width of the garden be \( w \) feet. The length is 3 feet more than twice the width, so: length \( l = 2w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) feet. The length is 3 feet more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 65 square feet: \[ l \times w = 65 \] Substitute \( l = 2w + 3 \): \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We can solve \( 2w^2 + 3w - 65 = 0 \) using the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here \( a = 2 \), \( b = 3 \), \( c = -65 \). First, compute the discriminant: \[ b^2 - 4ac = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{529} = 23 \] Now: \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Case 2: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) Since width cannot be negative, \( w = 5 \). --- **Step 6: Find length** \[ l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \] --- **Step 7: Check area** Area = \( 13 \times 5 = 65 \) square feet. Correct. --- **Final answer:** width = 5 feet, length = 13 feet.

6. A triangle has vertices at A(12, 5), B(18, 5), and C(15, 17). After a dilation centered at the origin with a scale factor of 2/3, followed by a reflection over the y-axis, which properties (angles, side lengths, parallelism) are preserved? List all that apply. Answer: angles and parallelism Solution: Apply dilation with scale factor 2/3 centered at origin. New coordinates: A'(8, 10/3), B'(12, 10/3), C'(10, 34/3). Side lengths are multiplied by 2/3, so distances change.
   Full step-by-step solution

   Step 1: Apply dilation with scale factor 2/3 centered at origin. New coordinates: A'(8, 10/3), B'(12, 10/3), C'(10, 34/3). Side lengths are multiplied by 2/3, so distances change. Angles remain the same because dilation is a similarity transformation. Step 2: Apply reflection over the y-axis. Coordinates become A''(-8, 10/3), B''(-12, 10/3), C''(-10, 34/3). Reflection preserves both distances and angles. Step 3: Combined effect: Dilation changes side lengths (not preserved), but angles are preserved by both transformations. Parallelism is preserved because both dilation and reflection preserve parallelism (lines remain lines, parallel lines stay parallel). Step 4: Properties preserved: angles and parallelism. Side lengths are not preserved because the dilation changed them. The answer is angles and parallelism.

7. A triangle with vertices at (12, 9), (12, 15), and (18, 9) is dilated by a factor of 2/3 with respect to the origin. After the dilation, which properties are preserved: angles, distances, and/or parallelism? Answer: Angles and parallelism are preserved; distances are not preserved. Solution: Identify the original triangle's side lengths. The vertices are (12,9), (12,15), and (18,9). The vertical side from (12,9) to (12,15) has length 6.
   Full step-by-step solution

   Step 1: Identify the original triangle's side lengths. The vertices are (12,9), (12,15), and (18,9). The vertical side from (12,9) to (12,15) has length 6. The horizontal side from (12,9) to (18,9) has length 6. The hypotenuse from (12,15) to (18,9) has length sqrt((18-12)^2 + (9-15)^2) = sqrt(6^2 + (-6)^2) = sqrt(36+36) = sqrt(72) = 6*sqrt(2). So original side lengths: 6, 6, 6*sqrt(2). Step 2: Apply the dilation factor of 2/3 to each vertex. Multiply each coordinate by 2/3: (12*2/3, 9*2/3) = (8, 6); (12*2/3, 15*2/3) = (8, 10); (18*2/3, 9*2/3) = (12, 6). Step 3: Find the new side lengths. Vertical side from (8,6) to (8,10): length = 4. Horizontal side from (8,6) to (12,6): length = 4. Hypotenuse from (8,10) to (12,6): sqrt((12-8)^2 + (6-10)^2) = sqrt(4^2 + (-4)^2) = sqrt(16+16) = sqrt(32) = 4*sqrt(2). Step 4: Compare original and new side lengths: 6 vs 4, 6 vs 4, 6*sqrt(2) vs 4*sqrt(2). The distances are multiplied by 2/3, so distances are NOT preserved. Step 5: Check angles. The original triangle has a right angle at (12,9) because sides are vertical and horizontal. The new triangle also has a right angle at (8,6) because sides are still vertical and horizontal. The other angles are the same because the ratio of sides is the same (1:1:sqrt(2) in both cases). So angles ARE preserved. Step 6: Check parallelism. In the original triangle, the vertical side is parallel to the y-axis, and the horizontal side is parallel to the x-axis. In the new triangle, the vertical side is still parallel to the y-axis, and the horizontal side is still parallel to the x-axis. So parallelism IS preserved. The answer is: Angles and parallelism are preserved; distances are not preserved.