Emma is designing a logo that starts as a triangle with vertices at A(1, 2), B(3, 2), and C(2, 5). She first rotates the triangle 90° counterclockwise about the origin. Then she translates the rotated triangle 4 units to the right and 3 units down. Finally, she reflects the translated triangle across the x-axis. What are the coordinates of vertex C after this sequence of transformations?Answer: ______________
Emma is designing a geometric pattern for her art project. She starts with a triangle that has vertices at A(1, 2), B(3, 2), and C(2, 4). She first rotates the triangle 90° counterclockwise about the origin. Then she translates the rotated triangle 2 units to the right and 3 units up. Finally, she reflects the translated triangle across the x-axis. What are the coordinates of vertex C after this sequence of transformations?Answer: ______________
A rectangular garden has a length of 15 meters and a width of 10 meters. The owner wants to create a path of uniform width around the entire garden, which will increase the total area to 266 square meters. What is the width of the path in meters?Answer: ______________
A triangle has vertices at A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. It is first reflected across the y-axis, then rotated 90° counterclockwise about the origin, and finally translated 3 units to the right and 2 units down. What are the coordinates of the final image of vertex C after this sequence of transformations?Answer: ______________
Emma is designing a logo that starts as a triangle with vertices at A(1, 2), B(3, 2), and C(2, 5). She first rotates the triangle 90° counterclockwise about the origin. Then she translates the rotated triangle 4 units to the right and 3 units down. Finally, she reflects the translated triangle across the x-axis. What are the coordinates of vertex C after this sequence of transformations?Answer: (1, -4) Solution: Start with original point C(2, 5) Rotate 90° counterclockwise about origin: (x, y) → (-y, x) C(2, 5) → C'(-5, 2) Translate 4 units right and 3 units down: (x, y) → (x+4, y-3) C'(-5, 2) → C''(-5+4, 2-3) = C''(-1, -1) C''(-1, -1) → C'''(-1, -(-1)) = C'''(-1, 1) The final coordinates of vertex C…Full step-by-step solution
Step 1: Start with original point C(2, 5)
Step 2: Rotate 90° counterclockwise about origin: (x, y) → (-y, x)
C(2, 5) → C'(-5, 2)
Step 3: Translate 4 units right and 3 units down: (x, y) → (x+4, y-3)
C'(-5, 2) → C''(-5+4, 2-3) = C''(-1, -1)
Step 4: Reflect across x-axis: (x, y) → (x, -y)
C''(-1, -1) → C'''(-1, -(-1)) = C'''(-1, 1)
The final coordinates of vertex C are (-1, 1).
Emma is designing a geometric pattern for her art project. She starts with a triangle that has vertices at A(1, 2), B(3, 2), and C(2, 4). She first rotates the triangle 90° counterclockwise about the origin. Then she translates the rotated triangle 2 units to the right and 3 units up. Finally, she reflects the translated triangle across the x-axis. What are the coordinates of vertex C after this sequence of transformations?Answer: (0, -1) Solution: Start with the original coordinates of point C: (2, 4) Apply the first transformation - rotate 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). So (2, 4) becomes (-4, 2).Full step-by-step solution
Step 1: Start with the original coordinates of point C: (2, 4)
Step 2: Apply the first transformation - rotate 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). So (2, 4) becomes (-4, 2).
Step 3: Apply the second transformation - translate 2 units right and 3 units up. This means add 2 to the x-coordinate and 3 to the y-coordinate. So (-4, 2) becomes (-4 + 2, 2 + 3) = (-2, 5).
Step 4: Apply the third transformation - reflect across the x-axis. The rule for this reflection is (x, y) → (x, -y). So (-2, 5) becomes (-2, -5).
The final coordinates of vertex C are (-2, -5).
A rectangular garden has a length of 15 meters and a width of 10 meters. The owner wants to create a path of uniform width around the entire garden, which will increase the total area to 266 square meters. What is the width of the path in meters?Answer: 2 Solution: Let x be the width of the path in meters. The path adds 2x to both the length and width (x on each side).Full step-by-step solution
Step 1: Let x be the width of the path in meters.
Step 2: The path adds 2x to both the length and width (x on each side).
Step 3: New length = 15 + 2x
Step 4: New width = 10 + 2x
Step 5: Total area with path = (15 + 2x)(10 + 2x) = 266
Step 6: Expand: 150 + 30x + 20x + 4x^2 = 266
Step 7: Simplify: 4x^2 + 50x + 150 = 266
Step 8: Subtract 266: 4x^2 + 50x - 116 = 0
Step 9: Divide by 2: 2x^2 + 25x - 58 = 0
Step 10: Factor: (2x + 29)(x - 2) = 0
Step 11: Solutions: x = -29/2 or x = 2
Step 12: Only x = 2 makes sense (width cannot be negative).
The width of the path is 2 meters.
A triangle has vertices at A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. It is first reflected across the y-axis, then rotated 90° counterclockwise about the origin, and finally translated 3 units to the right and 2 units down. What are the coordinates of the final image of vertex C after this sequence of transformations?Answer: (1, -5) Solution: Geometric transformations follow specific rules: reflection across the y-axis changes the x-coordinate's sign, rotation about the origin by 90° counterclockwise swaps coordinates and applies sign changes based on quadrant, and translation shifts coordinates by adding or subtracting values.Full step-by-step solution
Geometric transformations follow specific rules: reflection across the y-axis changes the x-coordinate's sign, rotation about the origin by 90° counterclockwise swaps coordinates and applies sign changes based on quadrant, and translation shifts coordinates by adding or subtracting values. Understanding these rules helps track how points move through transformation sequences.
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