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Congruence Concepts

Grade 8 · Geometry · Worksheet 1

  1. A triangle has vertices at A(1, 2), B(4, 2), and C(1, 5) on a coordinate plane. The triangle is first rotated 90° counterclockwise about the origin, then reflected across the x-axis. What are the coordinates of the final position of vertex C after both transformations? Answer: ______________
  2. Mere draws a quadrilateral on a coordinate plane with vertices at A(11, 2), B(19, 2), C(19, 10), and D(11, 10). She reflects the quadrilateral across the line y = x, and then translates the reflected shape 7 units to the left and 4 units down. What are the coordinates of the final position of vertex C (originally at (19, 10)) after both transformations? Answer: ______________
  3. A triangle with vertices at (5, -3), (8, 4), and (-2, 6) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (8, 4)? Answer: ______________
  4. A triangle has vertices at points A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. The triangle is reflected across the y-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of vertex C? Answer: ______________
  5. A square is drawn on a coordinate plane with vertices at (1, 1), (4, 1), (4, 4), and (1, 4). This square is first reflected across the line y = x, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of the vertex that was originally at (4, 1)? Answer: ______________
  6. Liam is designing a triangular garden plot. He places stakes at points A(2, 3), B(5, 7), and C(8, 3). His friend Noah wants to create a congruent triangular plot adjacent to it by applying a sequence of rigid motions: first a translation 4 units right and 2 units up, then a reflection across the vertical line x=9. What are the coordinates of the vertices of Noah's triangular garden plot after applying these transformations? Answer: ______________
  7. If triangle ABC is congruent to triangle DEF and AB = 5x - 3, DE = 2x + 9, find x. Answer: ______________
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Answer Key & Explanations

Congruence Concepts · Grade 8 · Worksheet 1

  1. A triangle has vertices at A(1, 2), B(4, 2), and C(1, 5) on a coordinate plane. The triangle is first rotated 90° counterclockwise about the origin, then reflected across the x-axis. What are the coordinates of the final position of vertex C after both transformations? Answer: (5,1) Solution: Start with the original coordinates of point C: C(1, 5) Apply the first transformation - rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x).
    Full step-by-step solution

    Step 1: Start with the original coordinates of point C: C(1, 5) Step 2: Apply the first transformation - rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) → (-y, x). For C(1, 5): x = 1, y = 5 After rotation: (-5, 1) Step 3: Apply the second transformation - reflection across the x-axis. The rule for this reflection is (x, y) → (x, -y). For the point (-5, 1): x = -5, y = 1 After reflection: (-5, -1) Step 4: Verify the final coordinates: (-5, -1) The answer is (-5, -1).

  2. Mere draws a quadrilateral on a coordinate plane with vertices at A(11, 2), B(19, 2), C(19, 10), and D(11, 10). She reflects the quadrilateral across the line y = x, and then translates the reflected shape 7 units to the left and 4 units down. What are the coordinates of the final position of vertex C (originally at (19, 10)) after both transformations? Answer: (3, 15) Solution: Start with the original point C at (19, 10). Reflect across the line y = x. This swaps the x- and y-coordinates: (19, 10) becomes (10, 19).
    Full step-by-step solution

    Step 1: Start with the original point C at (19, 10). Step 2: Reflect across the line y = x. This swaps the x- and y-coordinates: (19, 10) becomes (10, 19). Step 3: Apply the translation of 7 units left and 4 units down. Subtract 7 from the x-coordinate and subtract 4 from the y-coordinate: (10 - 7, 19 - 4) = (3, 15). The final coordinates of vertex C are (3, 15).

  3. A triangle with vertices at (5, -3), (8, 4), and (-2, 6) is reflected across the x-axis. What are the coordinates of the reflected vertex that was originally at (8, 4)? Answer: (8, -4) Solution: Identify the original coordinates of the vertex: (8, 4) Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged.
    Full step-by-step solution

    Step 1: Identify the original coordinates of the vertex: (8, 4) Step 2: Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate unchanged. Step 3: Apply the transformation: new x-coordinate = 8, new y-coordinate = -4 Step 4: The reflected vertex coordinates are (8, -4) The answer is (8, -4).

  4. A triangle has vertices at points A(2, 1), B(5, 1), and C(2, 4) on a coordinate plane. The triangle is reflected across the y-axis, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of vertex C? Answer: (4, 2) Solution: Geometric transformations follow specific rules. Reflection across the y-axis preserves the y-coordinate but changes the sign of the x-coordinate. Rotation about the origin by 90° counterclockwise swaps the coordinates and changes one sign.
    Full step-by-step solution

    Geometric transformations follow specific rules. Reflection across the y-axis preserves the y-coordinate but changes the sign of the x-coordinate. Rotation about the origin by 90° counterclockwise swaps the coordinates and changes one sign. When applying multiple transformations, the order matters - you apply them sequentially from the first to the last.

  5. A square is drawn on a coordinate plane with vertices at (1, 1), (4, 1), (4, 4), and (1, 4). This square is first reflected across the line y = x, then rotated 90° counterclockwise about the origin. What are the coordinates of the final position of the vertex that was originally at (4, 1)? Answer: (1, -4) Solution: Start with the original vertex at (4, 1). This swaps the x and y coordinates. So (4, 1) becomes (1, 4).
    Full step-by-step solution

    Step 1: Start with the original vertex at (4, 1). Step 2: Apply the first transformation: reflection across the line y = x. This swaps the x and y coordinates. So (4, 1) becomes (1, 4). Step 3: Apply the second transformation: rotation 90° counterclockwise about the origin. The rule for this rotation is (x, y) becomes (-y, x). So (1, 4) becomes (-4, 1). Step 4: The final coordinates are (-4, 1). The answer is (-4, 1).

  6. Liam is designing a triangular garden plot. He places stakes at points A(2, 3), B(5, 7), and C(8, 3). His friend Noah wants to create a congruent triangular plot adjacent to it by applying a sequence of rigid motions: first a translation 4 units right and 2 units up, then a reflection across the vertical line x=9. What are the coordinates of the vertices of Noah's triangular garden plot after applying these transformations? Answer: (10, 9), (8, 5), (6, 9) Solution: Rigid motions are transformations that preserve the size and shape of geometric figures, including translations, reflections, and rotations.
    Full step-by-step solution

    Rigid motions are transformations that preserve the size and shape of geometric figures, including translations, reflections, and rotations. When applying multiple transformations in sequence, each transformation affects all points of the figure. Translations move all points by the same distance in the same direction, while reflections create mirror images across a line of reflection. The order of transformations matters, as different sequences can produce different final positions even with the same individual transformations.

  7. If triangle ABC is congruent to triangle DEF and AB = 5x - 3, DE = 2x + 9, find x. Answer: 4 Solution: Since triangle ABC is congruent to triangle DEF, side AB corresponds to side DE, so AB = DE.
    Full step-by-step solution

    Step 1: Since triangle ABC is congruent to triangle DEF, side AB corresponds to side DE, so AB = DE. Step 2: Write the equation: 5x - 3 = 2x + 9 Step 3: Subtract 2x from both sides: 5x - 2x - 3 = 9 → 3x - 3 = 9 Step 4: Add 3 to both sides: 3x = 12 Step 5: Divide both sides by 3: x = 4 Step 6: Check: AB = 5(4) - 3 = 17, DE = 2(4) + 9 = 17. Both sides are equal. The answer is 4.