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Calculate the inclination of the line joining A(3, -1) and B(-2, -3). Calculate the inclination of the line joining F(4, 5) and G(4, 9). Calculate the gradient of a line with inclination 135 degrees.

Mathematics

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Here are the solutions for the requested problems.

1.a) Calculate the inclination of the line joining A(3, -1) and B(-2, -3).

  1. Calculate the gradient mm of the line.

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} m=−3−(−1)−2−3m = \frac{-3 - (-1)}{-2 - 3} m=−3+1−5m = \frac{-3 + 1}{-5} m=−2−5m = \frac{-2}{-5} m=25m = \frac{2}{5}
  2. Calculate the inclination θ\theta.

    tan⁡θ=m\tan \theta = m tan⁡θ=25\tan \theta = \frac{2}{5} θ=arctan⁡(25)\theta = \arctan\left(\frac{2}{5}\right) θ≈21.8014∘\theta \approx 21.8014^\circ

    Rounded to two decimal places:

    θ≈21.80∘\theta \approx 21.80^\circ

    The inclination is 21.80∘\boxed{\text{21.80}^\circ}.

    1.c) Calculate the inclination of the line joining F(4, 5) and G(4, 9).

  3. Calculate the gradient mm of the line.

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} m=9−54−4m = \frac{9 - 5}{4 - 4} m=40m = \frac{4}{0}

    The gradient is undefined.

  4. Determine the inclination θ\theta.

    An undefined gradient indicates a vertical line.
    The inclination of a vertical line is 90∘90^\circ.
    The inclination is 90∘\boxed{\text{90}^\circ}.

    2.a) Calculate the gradient of a line with inclination 135∘135^\circ.

  5. Use the formula m=tan⁡θm = \tan \theta.

    m=tan⁡(135∘)m = \tan(135^\circ) m=−1m = -1

    The gradient is −1\boxed{-1}.

    2.c) Calculate the gradient of a line with inclination 60∘60^\circ.

  6. Use the formula m=tan⁡θm = \tan \theta.

    m=tan⁡(60∘)m = \tan(60^\circ) m=3m = \sqrt{3}

    The gradient is 3\boxed{\sqrt{3}}.

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