Answer:
a) The length of segment AC is approximately 5.83 centimeters.
b) The angle ACD is approximately 34.5º.
Step-by-step explanation:
a) Since [tex]AB \perp BC[/tex], the length of segment [tex]AC[/tex] is determined by Pythagorean Theorem, that is:
[tex]AC = \sqrt{(5\,cm)^{2}+(3\,cm)^{2}}[/tex]
[tex]AC \approx 5.831\,cm[/tex]
The length of segment AC is approximately 5.831 centimeters.
b) Since [tex]AB \perp BC \perp AD[/tex], the length of segment [tex]AD[/tex] is determined by this Pythagorean identity:
[tex]AD = \sqrt{(3\,cm)^{2}+(5\,cm)^{2}+(4\,cm)^{2}}[/tex]
[tex]AD \approx 7.071\,cm[/tex]
The angle ACD is determined by the following trigonometric expression:
[tex]\cos C = \frac{AC}{CD}[/tex]
[tex]\cos C = \frac{5.831\,cm}{7.071\,cm}[/tex]
[tex]\cos C = 0.825[/tex]
[tex]C = \cos^{-1} 0.825[/tex]
[tex]C \approx 34.448^{\circ}[/tex]
The angle ACD is approximately 34.448º.
