The side length AC is 5.83 cm and angle ACD is 34.5 degrees
(a) Length AC
To do this, we make use of the following Pythagoras theorem in triangle ABC
[tex]\mathbf{AC^2 = AB^2 + BC^2}[/tex]
So, we have:
[tex]\mathbf{AC^2 = 3^2 + 5^2}[/tex]
[tex]\mathbf{AC^2 = 9 + 25}[/tex]
[tex]\mathbf{AC^2 = 34}[/tex]
Take square roots
[tex]\mathbf{AC = 5.83}[/tex]
(b) Angle ACD
To do this, we make use of the following tangent ratio
[tex]\mathbf{tan(C) = \frac{AD}{AC}}[/tex]
So, we have:
[tex]\mathbf{tan(C) = \frac{4}{5.83}}[/tex]
[tex]\mathbf{tan(C) = 0.6861}[/tex]
Take arc tan of both sides
[tex]\mathbf{C= 34.5}[/tex]
Hence, side length AC is 5.83 cm and angle ACD is 34.5 degrees
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