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Find the values of the following angles. (a) MLN (1mk) (b) LNP (2mks)

Mathematics

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Step-by-step solution

4 steps

Here are the values for the requested angles:

Part a· 2 steps

Find ∠MLN\angle MLN

  1. Identify the central angle subtended by arc MN.

    From the diagram, the angle at the center ∠LON=130∘\angle LON = 130^\circ and ∠LOM=50∘\angle LOM = 50^\circ.
    The angle ∠MON\angle MON is the difference between these two angles.

    ∠MON=∠LON−∠LOM=130∘−50∘=80∘\angle MON = \angle LON - \angle LOM = 130^\circ - 50^\circ = 80^\circ

  2. Apply the theorem that the angle at the circumference is half the angle at the center subtended by the same arc.

    Angle ∠MLN\angle MLN is an angle at the circumference subtended by arc MN.

    ∠MLN=12∠MON\angle MLN = \frac{1}{2} \angle MON ∠MLN=12×80∘\angle MLN = \frac{1}{2} \times 80^\circ ∠MLN=40∘\angle MLN = 40^\circ

    The value of ∠MLN\angle MLN is 40∘\boxed{40^\circ}.

Part b· 2 steps

Find ∠LNP\angle LNP

  1. Recognize that NP is a tangent to the circle at point N, and ON is a radius.

    The radius to the point of tangency is perpendicular to the tangent.
    Therefore, ∠ONP=90∘\angle ONP = 90^\circ.

  2. Find angle ∠ONL\angle ONL.

    In $\

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