NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Congruence and similarity

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Triangle ABC has vertices at A(0, 0), B(6, 0), and C(0, 8). Triangle DEF has vertices at D(0, 0), E(6, 0), and F(0, 8).

After a reflection over the y-axis followed by a translation of 4 units right and 3 units up, triangle DEF maps exactly onto triangle ABC. Determine the exact length of side AC in triangle ABC.

Provide your answer as a single exact value.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Triangle ABC has vertices A(0, 0), B(6, 0), and C(0, 8). The side AC is the distance from A to C.
  2. The coordinates of A are (0, 0) and C are (0, 8). Since both points lie on the y-axis, the distance between them is the absolute difference in their y-coordinates.
  3. Use the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). Substitute x1 = 0, y1 = 0, x2 = 0, y2 = 8.
  4. Compute (x2 - x1)^2 = (0 - 0)^2 = 0.
  5. Compute (y2 - y1)^2 = (8 - 0)^2 = 64.
  6. Add the squares: 0 + 64 = 64.
  7. Take the square root: sqrt(64) = 8.
  8. The length of side AC is 8 units.
  9. The transformation described (reflection over y-axis, then translation) maps DEF onto ABC, but since ABC and DEF have identical coordinates, they are congruent by definition. This confirms the side lengths are preserved.
  10. No further calculation is needed because AC is vertical and spans from (0,0) to (0,8), so length is 8.
  11. The length of side AC is exactly 8 units.
  12. Final verification: distance between (0,0) and (0,8) is |8 - 0| = 8. Correct.

Reference answer: 8

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