Q1: Which transformation moves every point of a figure the same distance in the same direction? (a) Rotation (b) Reflection (c) Translation (d) Dilation
Solution:
Ans: (c) Explanation: A translation slides a figure from one position to another without changing its size, shape, or orientation. Every point moves the same distance in the same direction. Rotation turns a figure around a point, reflection flips it over a line, and dilation changes its size.
Q2: A point \(A(3, 5)\) is reflected across the x-axis. What are the coordinates of the image point \(A'\)? (a) \((3, -5)\) (b) \((-3, 5)\) (c) \((-3, -5)\) (d) \((5, 3)\)
Solution:
Ans: (a) Explanation: When reflecting across the x-axis, the x-coordinate stays the same and the y-coordinate changes sign. Therefore, \(A(3, 5)\) becomes \(A'(3, -5)\). Option (b) would be a reflection across the y-axis, and option (c) would be a reflection across both axes.
Q3: A triangle is rotated \(90°\) counterclockwise about the origin. If a vertex is at \((4, 2)\), what are the coordinates of this vertex after the rotation? (a) \((2, 4)\) (b) \((-2, 4)\) (c) \((4, -2)\) (d) \((-4, -2)\)
Solution:
Ans: (b) Explanation: For a 90° counterclockwise rotation about the origin, the rule is \((x, y) \rightarrow (-y, x)\). Applying this to \((4, 2)\): the new coordinates are \((-2, 4)\). Option (a) represents a different transformation, and options (c) and (d) do not follow the rotation rule.
Q4: Which type of transformation preserves both the size and shape of a figure? (a) Dilation only (b) Translation, rotation, and reflection (c) Dilation and translation (d) Rotation only
Solution:
Ans: (b) Explanation:Rigid transformations (also called isometries) preserve both size and shape. These include translation, rotation, and reflection. A dilation changes the size of a figure, so it is not a rigid transformation. Therefore, option (b) is correct.
Q5: A figure is translated using the rule \((x, y) \rightarrow (x + 5, y - 3)\). If point \(B\) is at \((-2, 7)\), where is point \(B'\)? (a) \((3, 4)\) (b) \((-7, 10)\) (c) \((3, 10)\) (d) \((-2, 4)\)
Solution:
Ans: (a) Explanation: Apply the translation rule \((x, y) \rightarrow (x + 5, y - 3)\) to point \(B(-2, 7)\): \(x' = -2 + 5 = 3\) \(y' = 7 - 3 = 4\) Therefore, \(B'(3, 4)\). The other options result from incorrect application of the translation rule.
Q6: What is the image of point \(C(6, -4)\) after a reflection across the y-axis? (a) \((-6, -4)\) (b) \((6, 4)\) (c) \((-6, 4)\) (d) \((-4, 6)\)
Solution:
Ans: (a) Explanation: When reflecting across the y-axis, the y-coordinate remains the same and the x-coordinate changes sign. Thus, \(C(6, -4)\) becomes \(C'(-6, -4)\). Option (b) is a reflection across the x-axis, and option (c) is a reflection across both axes.
Q7: A quadrilateral is rotated \(180°\) about the origin. What is the image of vertex \(D(5, -3)\)? (a) \((5, 3)\) (b) \((-5, 3)\) (c) \((-5, -3)\) (d) \((3, -5)\)
Solution:
Ans: (b) Explanation: For a 180° rotation about the origin, the rule is \((x, y) \rightarrow (-x, -y)\). Applying this to \(D(5, -3)\): \(x' = -5\) \(y' = -(-3) = 3\) Therefore, \(D'(-5, 3)\). The other options do not follow the 180° rotation rule.
Q8: Which transformation would map point \(P(2, 8)\) to \(P'(2, -8)\)? (a) Translation 16 units down (b) Reflection across the x-axis (c) Reflection across the y-axis (d) Rotation \(90°\) clockwise
Solution:
Ans: (b) Explanation: The x-coordinate remains 2 while the y-coordinate changes from 8 to -8, which means the sign of y changed. This is the characteristic of a reflection across the x-axis. Option (a) would give \((2, -8)\) but through translation, not reflection. Options (c) and (d) produce different results.
## Section B: Fill in the Blanks Q9:A transformation that flips a figure over a line is called a(n) __________.
Solution:
Ans: reflection Explanation: A reflection is a transformation that produces a mirror image of a figure across a line called the line of reflection.
Q10:When a figure is rotated, the fixed point around which the figure turns is called the __________.
Solution:
Ans: center of rotation Explanation: The center of rotation is the point that remains fixed during a rotation. All other points in the figure rotate around this center point.
Q11:The rule for reflecting a point across the x-axis is \((x, y) \rightarrow (x, __________)\).
Solution:
Ans: -y Explanation: When reflecting across the x-axis, the x-coordinate stays the same and the y-coordinate becomes its opposite. Thus, the rule is \((x, y) \rightarrow (x, -y)\).
Q12:Transformations that preserve the size and shape of a figure are called __________ transformations.
Solution:
Ans: rigid (or isometries) Explanation:Rigid transformations, also known as isometries, include translations, rotations, and reflections. These transformations do not change the dimensions or angles of the figure.
Q13:A translation of 4 units right and 7 units up can be written as the rule \((x, y) \rightarrow (x + 4, y + __________)\).
Solution:
Ans: 7 Explanation: In a translation, moving right adds to the x-coordinate and moving up adds to the y-coordinate. Since we move 7 units up, we add 7 to y, giving \((x, y) \rightarrow (x + 4, y + 7)\).
Q14:The rule for rotating a point \(90°\) counterclockwise about the origin is \((x, y) \rightarrow (__________, x)\).
Solution:
Ans: -y Explanation: For a 90° counterclockwise rotation about the origin, the transformation rule is \((x, y) \rightarrow (-y, x)\). The x-coordinate becomes the new y-coordinate, and the y-coordinate becomes the negative of the new x-coordinate.
## Section C: Word Problems Q15:A designer creates a logo on a coordinate plane with one vertex at point \(A(3, 6)\). She translates the logo 5 units left and 2 units down. What are the coordinates of the image of point \(A\) after the translation?
Solution:
Ans: Step 1: Moving 5 units left means subtracting 5 from the x-coordinate: \(3 - 5 = -2\) Step 2: Moving 2 units down means subtracting 2 from the y-coordinate: \(6 - 2 = 4\) Final Answer: \(A'(-2, 4)\)
Q16:A triangle has vertices at \(P(2, 3)\), \(Q(5, 3)\), and \(R(4, 7)\). The triangle is reflected across the y-axis. What are the coordinates of the three vertices of the reflected triangle?
Solution:
Ans: Step 1: For reflection across the y-axis, use the rule \((x, y) \rightarrow (-x, y)\) Step 2: Apply to each vertex: \(P(2, 3) \rightarrow P'(-2, 3)\) \(Q(5, 3) \rightarrow Q'(-5, 3)\) \(R(4, 7) \rightarrow R'(-4, 7)\) Final Answer: \(P'(-2, 3)\), \(Q'(-5, 3)\), \(R'(-4, 7)\)
Q17:A square on a coordinate grid has one vertex at \(S(-3, 2)\). The square is rotated \(90°\) counterclockwise about the origin. What are the coordinates of the image of vertex \(S\)?
Solution:
Ans: Step 1: Use the rule for 90° counterclockwise rotation: \((x, y) \rightarrow (-y, x)\) Step 2: Apply to \(S(-3, 2)\): \(x' = -y = -2\) \(y' = x = -3\) Final Answer: \(S'(-2, -3)\)
Q18:A park designer places a fountain at point \(F(8, -5)\) on a coordinate map. The design is rotated \(180°\) about the origin to create a symmetric layout. What are the new coordinates of the fountain after the rotation?
Solution:
Ans: Step 1: Use the rule for 180° rotation: \((x, y) \rightarrow (-x, -y)\) Step 2: Apply to \(F(8, -5)\): \(x' = -8\) \(y' = -(-5) = 5\) Final Answer: \(F'(-8, 5)\)
Q19:A game developer translates a character sprite from position \(C(-4, 9)\) using the rule \((x, y) \rightarrow (x + 7, y - 6)\). After the translation, the sprite is reflected across the x-axis. What are the final coordinates of the sprite?
Solution:
Ans: Step 1: Apply translation rule to \(C(-4, 9)\): \(x' = -4 + 7 = 3\) \(y' = 9 - 6 = 3\) Position after translation: \((3, 3)\) Step 2: Reflect across x-axis using \((x, y) \rightarrow (x, -y)\): \(x'' = 3\) \(y'' = -3\) Final Answer: \((3, -3)\)
Q20:An architect designs a building feature with a corner at point \(T(6, 4)\). The design is first reflected across the y-axis, then translated 3 units up. What are the final coordinates of point \(T\) after both transformations?
Solution:
Ans: Step 1: Reflect across y-axis using \((x, y) \rightarrow (-x, y)\): \(T(6, 4) \rightarrow T'(-6, 4)\) Step 2: Translate 3 units up: add 3 to y-coordinate: \(x'' = -6\) \(y'' = 4 + 3 = 7\) Final Answer: \(T''(-6, 7)\)
The document Worksheet (with Solutions): Performing Transformations is a part of the Grade 9 Course Integrated Math 1.
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