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Find the angle CKD given the coordinates of A, C, D and the equation of line BD.

Mathematics

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Step-by-step solution

3 steps

To determine the size of ∠CKD\angle CKD, you'll need to find the gradients of the lines CK and KD.

  1. Find the coordinates of point K.

    • K is the intersection of the diagonals AC and BD.
    • From question 3.2, you should have calculated the gradient of AC. Use this gradient and one of the points A(−3,−8)(-3, -8) or C(3,4)(3, 4) to find the equation of line AC.
    • The equation of line BD is given as y=−43x−2y = -\frac{4}{3}x - 2.
    • Solve the equations for line AC and line BD simultaneously to find the coordinates of K.

  2. Calculate the gradients of CK and KD.

    • Use the coordinates of C(3,4)(3, 4), K (from Step 1), and D(3,−6)(3, -6) to find mCKm_{CK} and mKDm_{KD}.

  3. Use the formula for the angle between two lines.

    • If m1m_1 and m2m_2 are the gradients of two lines, the angle θ\theta between them is given by:
    tan⁡θ=∣m1−m21+m1m2∣\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|
    • Substitute mCKm_{CK} and mKDm_{KD} into this formula to find ∠CKD\angle CKD.
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