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Triangle PQR has vertices P(5, -1), Q(-2, -8) and R(13, 3) - Scottish Highers Maths - Question 1 - 2023

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Triangle PQR has vertices P(5, -1), Q(-2, -8) and R(13, 3). (a) Find the equation of the altitude from P. (b) Calculate the angle that the side PR makes with the p... show full transcript

Worked Solution & Example Answer:Triangle PQR has vertices P(5, -1), Q(-2, -8) and R(13, 3) - Scottish Highers Maths - Question 1 - 2023

Step 1

Find the equation of the altitude from P.

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Answer

To find the equation of the altitude from point P, we first need to determine the gradient of line QR.

  1. Determine the coordinates of points Q and R:

    • Q = (-2, -8)
    • R = (13, 3)

  2. Calculate the gradient (m) of line QR: Using the formula for the gradient: mQR=y2−y1x2−x1=3−(−8)13−(−2)=3+813+2=1115m_{QR} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-8)}{13 - (-2)} = \frac{3 + 8}{13 + 2} = \frac{11}{15}

  3. Find the perpendicular gradient: The gradient of the perpendicular altitude from P is the negative reciprocal of the gradient of QR: maltitude=−1mQR=−1511m_{altitude} = -\frac{1}{m_{QR}} = -\frac{15}{11}

  4. Use point P to write the equation: The coordinates of P are (5, -1). We can use the point-slope form of a linear equation: y−y1=m(x−x1)y - y_1 = m(x - x_1) Plugging in our values: y−(−1)=−1511(x−5)y - (-1) = -\frac{15}{11}(x - 5) Simplifying this gives us the altitude equation: y+1=−1511x+7511y + 1 = -\frac{15}{11}x + \frac{75}{11} Therefore, the equation of the altitude is: y=−1511x+7511−1=−1511x+6411y = -\frac{15}{11}x + \frac{75}{11} - 1 = -\frac{15}{11}x + \frac{64}{11}

Step 2

Calculate the angle that the side PR makes with the positive direction of the x-axis.

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Answer

To determine the angle that side PR makes with the positive direction of the x-axis, we follow these steps:

  1. Determine the coordinates of points P and R:

    • P = (5, -1)
    • R = (13, 3)

  2. Calculate the gradient (m) of line PR: Using the gradient formula: mPR=y2−y1x2−x1=3−(−1)13−5=3+113−5=48=12m_{PR} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-1)}{13 - 5} = \frac{3 + 1}{13 - 5} = \frac{4}{8} = \frac{1}{2}

  3. Calculate the angle (θ) using the tangent function: The angle can be found using the formula: tan⁡(θ)=mPR\tan(\theta) = m_{PR} Hence: θ=tan⁡−1(12)\theta = \tan^{-1}(\frac{1}{2})

  4. Calculate the angle in degrees: Using a calculator: θ≈26.57∘\theta \approx 26.57^{\circ} Rounding this gives us: θ≈27∘\theta \approx 27^{\circ} Thus, the angle that side PR makes with the positive direction of the x-axis is approximately 27 degrees.

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