NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Algebraic expressions and equations

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Required skills (click to reveal)

A craft designer is creating a wooden frame with a diagonal support. The horizontal side is 68 cm and the vertical side is 91 cm.

Using the Pythagorean theorem and algebraic simplification, find the exact length of the diagonal support in centimetres. Express your answer as a simplified radical expression.

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Combining like terms as equal tiles. Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.xxxxx3x2x3x + 2x = 5x
Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.
Distributive law as a rectangle. A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.x433x123(x + 4) = 3x + 12
A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The wooden frame forms a right-angled triangle with the horizontal side (68 cm) and vertical side (91 cm). The diagonal support is the hypotenuse, which must be found using the Pythagorean theorem.
  2. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b): c² = a² + b².
  3. Substitute the given side lengths: a = 68 cm, b = 91 cm. So, c² = 68² + 91².
  4. Calculate 68²: 68 × 68 = 4624.
  5. Calculate 91²: 91 × 91 = 8281.
  6. Add the squares: 4624 + 8281 = 12905.
  7. Now find c = √12905. Factor 12905 to simplify the radical.
  8. Check for square factors in 12905. Start by testing divisibility by small primes. 12905 is odd, so not divisible by 2. Sum of digits: 1+2+9+0+5 = 17, not divisible by 3. Ends in 5, so divisible by 5: 12905 ÷ 5 = 2581.
  9. Now factor 2581. Test divisibility: not by 2, 3, 5. Try 7: 2581 ÷ 7 ≈ 368.7, not integer. Try 11: 2 - 5 + 8 - 1 = 4, not divisible by 11. Try 13: 2581 ÷ 13 ≈ 198.5, not integer. Try 17: 2581 ÷ 17 = 151.82, no. Try 19: 2581 ÷ 19 = 135.84, no. Try 23: 2581 ÷ 23 = 112.217, no. Try 29: 2581 ÷ 29 = 89. So 2581 = 29 × 89.
  10. So 12905 = 5 × 29 × 89. None of these factors are repeated, so no perfect square factors other than 1. Therefore, √12905 cannot be simplified further.
  11. Thus, the exact length of the diagonal support is √12905 cm.
  12. Check: 113² = 12769, 114² = 12996. 12905 is between them, so √12905 is not an integer, confirming that the radical is already simplified.

Reference answer: sqrt(12905) cm

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